Charge Transport, Configuration Interaction and
Rydberg States under Density Functional Theory
by
Chiao-Lun Cheng
B.S., Univ. of California, Berkeley (2003)
B.A., Univ. of California, Berkeley (2003)
Submitted to the Department of Chemistry
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
September 2008
c Massachusetts Institute of Technology 2008. All rights reserved.
Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Department of Chemistry
Aug 15, 2008
Certified by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Troy Van Voorhis
Associate Professor of Chemistry
Thesis Supervisor
Accepted by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Robert W. Field
Chairman, Department Committee on Graduate Students
This doctoral thesis has been examined by a Committee of the Department of Chemistry as follows:
Professor Jianshu Cao . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Chairman, Thesis Committee
Associate Professor of Chemistry
Professor Troy Van Voorhis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Thesis Supervisor
Associate Professor of Chemistry
Professor Robert W. Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Member, Thesis Committee
Haslam and Dewey Professor of Chemistry
2
Charge Transport, Configuration Interaction and Rydberg
States under Density Functional Theory
by
Chiao-Lun Cheng
Submitted to the Department of Chemistry
on Aug 15, 2008, in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy
Abstract
Density functional theory (DFT) is a computationally efficient formalism for studying
electronic structure and dynamics. In this work, we develop DFT-based excitedstate methods to study electron transport, Rydberg excited states and to characterize
diabatic electronic configurations and couplings. We simulate electron transport in a
molecular wire using real-time time-dependent density functional theory in order to
study the conduction of the wire. We also use constrained density functional theory to
obtain diabatic states and diabatic couplings, and use these excited-state properties
in a configuration-interaction method that treats both dynamic and static correlation.
Lastly, we use eDFT, an excited-state self-consistent-field method, to determine the
energies of excited Rydberg atomic states.
Thesis Supervisor: Troy Van Voorhis
Title: Associate Professor of Chemistry
3
4
Acknowledgments
I thank my advisor Prof. Troy Van Voorhis for his leadership and guidance. His
door was always open when I had questions or ideas (however silly and impractical)
I needed advice on, and his broad knowledge base and endless patience as a mentor
has had a huge impact on my learning the past five years. I am especially thankful
to Troy for giving me the independence to pursue my own ideas, even with their
dismal success rate. This exploration space has helped me develop as a scientist and
a thinker over my years of graduate study.
I would like to thank my thesis committee members Profs. Robert W. Field and
Jianshu Cao for their advice. I thank my undergraduate physical chemistry lecturer
Prof. Max Wolfsberg for starting me on this journey - without those early discussions
I would not have chosen this path. I thank my undergraduate research advisors Profs.
Paul A. Bartlett and Judith P. Klinman for introducing me to the world of research.
I would especially like to thank my Klinman group mentor Matthew P. Meyer for
showing me what an interesting place the world can be.
Fellow physical chemistry colleagues have helped make my five years very enjoyable. I thank Jim Witkoskie, Steve Presse, Qin Wu, Xiaogeng Song, Jeremy
Evans, Seth Difley and Steve Coy for many interesting and stimulating discussions.
I thank Ziad Ganim for his assistance in figuring out CHARMM. I would also like to
thank Aiyan Lu, Yuan-Chung Cheng, Shilong Yang, Xiang Xia, Jianlan Wu, Serhan
Altunata, Vassiliy Lubchenko, Leeping Wang, Tim Kowalczyk and Ben Kaduk for
maintaining the spirit of camaraderie I’ve felt in the “zoo”.
I thank friends in Salon Chaos, the MIT Aikido Club, MIT ROCSA and the MIT
Singapore Students Society - they have been an invaluable source of warmth and
sanity. Salon Chaos offered a truly unique forum for exchanging ideas, I would like
to thank Mo-Han Hsieh and I-Fan Lin for organizing it. I thank my neighbors Emma
Chung and her housemates for providing me with a second home during my last few
years of graduate school. I thank Rebecca Somers for her perspective during stressful
times, and Huiyi Chen for always being there.
5
I thank my sister Ting-Yen for her encouraging words throughout the years. Most
of all, I thank my parents, a constant source of support throughout my life, for
granting me the freedom to pursue my dreams. This dissertation is dedicated to
them.
6
To my parents
7
8
Contents
1 Introduction
1.1
1.2
1.3
19
The Many-Body Schroedinger Equation . . . . . . . . . . . . . . . . .
19
1.1.1
Hartree-Fock . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
1.1.2
Electronic Correlation . . . . . . . . . . . . . . . . . . . . . .
26
Density Functional Theory . . . . . . . . . . . . . . . . . . . . . . . .
30
1.2.1
The Hohenberg-Kohn Theorems . . . . . . . . . . . . . . . . .
30
1.2.2
The Kohn-Sham Formalism . . . . . . . . . . . . . . . . . . .
32
1.2.3
Constrained Density Functional Theory (CDFT) . . . . . . . .
34
1.2.4
Time-Dependent Density Functional Theory (TDDFT) . . . .
37
1.2.5
Approximate Exchange-Correlation Potentials . . . . . . . . .
40
Structure of this Dissertation . . . . . . . . . . . . . . . . . . . . . .
47
2 Molecular Conductance using
Real-Time Time-Dependent
Density Functional Theory
49
2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
2.1.1
The Landauer-Büttiker Model . . . . . . . . . . . . . . . . . .
49
2.1.2
Non-Equilibrium Green’s Function (NEGF) Methods . . . . .
50
2.1.3
Time-Dependent Density Functional Theory . . . . . . . . . .
51
Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
2.2.1
Integrating the TDKS equations . . . . . . . . . . . . . . . . .
53
2.2.2
Numerical Validation . . . . . . . . . . . . . . . . . . . . . . .
59
Application: Conductance of a molecular wire . . . . . . . . . . . . .
61
2.2
2.3
9
2.4
2.3.1
Voltage Definitions . . . . . . . . . . . . . . . . . . . . . . . .
62
2.3.2
Current Averaging . . . . . . . . . . . . . . . . . . . . . . . .
65
2.3.3
Comparison to NEGF results . . . . . . . . . . . . . . . . . .
68
2.3.4
Voltage Biased Case . . . . . . . . . . . . . . . . . . . . . . .
70
2.3.5
Transient Fluctuations . . . . . . . . . . . . . . . . . . . . . .
71
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
74
3 Dissociation Curves using
Constrained Density Functional Theory
Configuration Interaction
77
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
77
3.2
State of the Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
78
3.3
Our Method: Constrained Density Functional Theory Configuration
Interaction (CDFT-CI) . . . . . . . . . . . . . . . . . . . . . . . . . .
79
3.3.1
Identifying an Active Space . . . . . . . . . . . . . . . . . . .
80
3.3.2
Computing Matrix Elements . . . . . . . . . . . . . . . . . . .
81
Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
85
3.4.1
H+
2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
85
3.4.2
H2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
87
3.4.3
LiF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
88
3.5
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
91
3.6
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . .
92
3.4
4 Rydberg Energies using Excited-State DFT
93
4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
93
4.2
Ground State Kohn-Sham DFT . . . . . . . . . . . . . . . . . . . . .
95
4.3
Excited State Kohn-Sham DFT . . . . . . . . . . . . . . . . . . . . .
96
4.4
State of the Art: Linear Response TDDFT . . . . . . . . . . . . . . .
97
4.5
Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
98
4.5.1
The Excited-State Exchange-Correlation Potential . . . . . . .
98
4.5.2
Numerical Evaluation of Rydberg Energies . . . . . . . . . . . 103
10
4.6
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.6.1
4.7
Extrema of the ground state energy functional . . . . . . . . . 106
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5 Conclusion
113
A Magnus and Runge-Kutta Munthe-Kaas Integrators
115
A.1 Magnus Integrators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
A.2 RKMK Integrators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
11
12
List of Figures
1-1 UHF orbital filling - ↑ are α-electrons, ↓ are β-electrons. Each horizontal line represents a different orbital, and the α and β electrons fill
different orbitals in an Aufbau manner, from the bottom up. In this
particular example, there are 4 α electrons, 4 β electrons and 6 basis
functions forming 12 spin-orbitals . . . . . . . . . . . . . . . . . . . .
25
1-2 Typical Algorithm for Solving the HF equations . . . . . . . . . . . .
26
1-3 Some CASSCF(2,3) configurations
. . . . . . . . . . . . . . . . . . .
29
1-4 Algorithm for Solving the CDFT equations . . . . . . . . . . . . . . .
36
1-5 H+
2 Energies in cc-PVTZ basis . . . . . . . . . . . . . . . . . . . . . .
45
1-6 H2 Energies in cc-PVTZ basis . . . . . . . . . . . . . . . . . . . . . .
46
2-1 Predictor-Corrector routine for the 2nd order Magnus integrator. The
order row shows the time order (in dt) to which the matrices in the
same column are correct to. . . . . . . . . . . . . . . . . . . . . . . .
58
2-2 Minimum wall time required to obtain a prescribed average absolute
error in the final density matrix of methane (B3LYP/6-31G*, 120
a.u of propagation) using various approximate propagators: 2nd order
Magnus (green dashed), 4th order Runge-Kutta (teal dot-dashed with
squares), 4th order Magnus (red solid) and 6th order Magnus (blue
dotted). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
60
2-3 Schematic of the source-wire-drain geometry used in the present simulations. The bias is applied to the left and right groups of atoms, which
act as a source and drain for electrons, respectively. For different wire
lengths (e.g. 50 carbons versus 100) the wire length is kept fixed and
the size of the source and sink are varied. . . . . . . . . . . . . . . . .
61
2-4 Initial density corresponding to a chemical potential bias in polyacetylene. Red indicates charge accumulation and green charge depletion
relative to the unbiased ground state. At time t=0 the bias is removed
and current flows from left to right. . . . . . . . . . . . . . . . . . . .
65
2-5 Transient current through the central four carbons in C50 H52 at a series
of different chemical potential biases. There is an increase in current
as voltage is increased, along with large, persistent fluctuations in the
current. The currents are converged with respect to time step and
the apparent noise is a result of physical fluctuations in particle flow
through the wire. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
65
2-6 Transient current through the central four carbons in C50 H52 at a series
of different chemical potential biases smoothed over a time window of
width ∆t = .36f s. The average currents are now move clearly visible.
The slow decay of the current at later times results from the partial
equilibration of the finite left and right leads. . . . . . . . . . . . . . .
66
2-7 Maximum smoothed current through the central four carbons in C50 H52
as a function of chemical potential bias (red pluses). For comparison,
we also present the analogous result for the central carbons in C100 H102
(green squares) demonstrating convergence of the calculation with respect to lead size. The blue line is a linear fit to the C50 H52 data
at low bias indicating that polyacetylene is an Ohmic resistor with a
conductance of ≈ .8G0 . . . . . . . . . . . . . . . . . . . . . . . . . .
14
67
2-8 Maximum smoothed current through the central four carbons in C50 H52
as a function of chemical potential bias using real-time TDDFT (top
red line) and an NEGF approach described in the text (bottom blue
line). The two calculations are nearly identical at low bias and differ
somewhat at higher biases due to the lack of self-consistency in the
NEGF results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
2-9 Maximum smoothed current through the central four carbons in C50 H52
as a function of chemical potential bias (top red line) and voltage bias
(bottom green line). The results are quite similar until 4 V, at which
point the bias is so large that the finite width of the valence band for
polyacetylene causes the conductance to plateau in the voltage biased
case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
70
2-10 Transient current through the central four carbons in C100 H102 at a
series of different chemical potential biases. The current fluctuations
previously observed with smaller reservoirs in C50 H52 persist and are
therefore not associated with a finite size effect. . . . . . . . . . . . .
71
2-11 Statistical noise in the current through the central four carbons in
C100 H102 . The data (squares) can be fit to a sub-Poissonian distribution
(line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
72
3-1 Figure taken from Ref. [1] Obtaining effective constraint value from
promolecular densities. The constraints are on Fragment A. Integration
over the striped area will give NCeff . . . . . . . . . . . . . . . . . . . .
83
3-2 Figure taken from Ref. [1] Potential curves of H+
2. . . . . . . . . . . .
86
3-3 Figure taken from Ref. [1] Potential curves of H2 . . . . . . . . . . . .
88
3-4 Figure taken from Ref. [1] Error comparison of UB3LYP and CB3LYPCI. Errors are calculated against full CI results. . . . . . . . . . . . .
15
89
3-5 Figure taken from Ref. [1] Singlet potential curves of LiF by B3LYP
and CB3LYP-CI as compared to the triplet curve by B3LYP. The Ms =0
curve of B3LYP is a broken-symmetry solution at large R and has a
positive charge of 0.3 on Li at the dissociation limit. . . . . . . . . . .
90
3-6 Figure taken from Ref. [1] CB3LYP-CI(S), CB3LYP-CI and CBLYPCI potential curves as compared to OD(2). CB3LYP-CI(S) is the
CB3LYP-CI using stockholder weights.
. . . . . . . . . . . . . . . .
91
3-7 Figure taken from Ref. [1] Weights of configurations in the final ground
state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
92
4-1 Hydrogen atom excited state exchange potentials (eV vs. Angstroms).
The solid red line shows the exact exchange potential and the subsequent lines (moving upward at the origin) show the 1s, 2s and 3s LSDA
exchange potentials from eDFT using the exact density. While none of
the potentials individually has the correct decay, subsequent potentials
decay more and more slowly. . . . . . . . . . . . . . . . . . . . . . . . 100
4-2 Exchange potentials for hydrogen. The dotted green line is the groundstate LSDA exchange potential [2]. The solid red line is the exact exchange potential. The blue crosses are the excited-state density LSDA
exchange potential at the classical turning points r = 2n2 . . . . . . . 102
A-1 Extrapolation routine for the 4th order Magnus integrator. The order
row shows the time order (in dt) to which the matrices in the same
column are correct to. . . . . . . . . . . . . . . . . . . . . . . . . . . 116
16
List of Tables
4.1
eDFT orbital eigenvalues of the hydrogen atom computed using LSDA
exchange and the exact densities. Each row corresponds to a particular
excited state, and the columns give the appropriate orbital eigenvalues. Note that, as one occupies more and more diffuse orbitals, the 1s
orbital eigenvalue approaches the correct value of -13.6 eV, naturally
correcting for the poor LSDA exchange-only estimate of the ionization
energy of ǫ1s =-6.3 eV. . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2
99
Hydrogen Atom eDFT Excitation Energies (eV). Italics indicate difference from the exact energy. NC indicates no converged state was
found. Average and Root Mean Square errors are also indicated. . . . 103
4.3
Lithium Atom Excitation Energies (eV). Italics indicate difference
from the experimental energy. ELDA−X and EB3LY P denote eDFT energies with the appropriate functional, while TDLDA and TDB3LYP
are the corresponding TDDFT energies. CS00 [4] contains an asymptotic corrections to the B3LYP exchange potential. In the TDDFT
columns NA indicates energy levels above the ionization energy. Experimental numbers from Ref. [5]. . . . . . . . . . . . . . . . . . . . . 105
17
18
Chapter 1
Introduction
In this chapter, we introduce some of the fundamental laws and approaches to electronic structure, which will set the background and notation for the work in this
dissertation. We end by describing how the rest of this dissertation is structured.
1.1
The Many-Body Schroedinger Equation
Here we introduce the Schroedinger equation, assuming non-relativistic electrons, i.e.
the kinetic energy is small compared to the rest mass of the electron, and also immobile nuclei, i.e. the Bohn-Oppenheimer approximation and no magnetic interactions.
The electronic Hamiltonian H is a Hermitian operator with units of energy, and
can be expressed as the sum of several parts:
H = T + vext + vee
X 1
T =
− ∇2i
2
i
X
vext =
vext (ri )
(1.1)
i
vee =
X
i<j
1
|ri − rj |
T is the kinetic energy operator, and is the sum of kinetic energy operators for
each electron coordinate ri . vext is the external potential, which is the single-particle
external potential summed over all the electron coordinates. The single-particle ex19
ternal potential is a function which maps a single 3-d electron coordinate in length
units to an energy, vext (r). It represents the electric potential experienced by electrons, which in molecular systems originates from the nuclei. vee is the Coulomb
repulsion between electrons, and is a sum of the reciprocal distance between all pairs
of electron positions.
A given electronic system is completely specified by the function vext , and has a
state specified by a wavefunction Ψ(x1 , x2 , · · · , xN ) where N is the number of electrons
and xi include both spin and spatial coordinates.
Assuming that Ψ is normalized, the energy of a system in the state Ψ is obtained
by taking its expectation with respect to the Hamiltonian H.
E=
Z
Ψ† (~r)HΨ(~r)d~r
We now extremize the energy of an electronic system while enforcing the constraint
of normalizability using the Lagrange multiplier λ
W =
Z
†
Ψ (~r)HΨ(~r)d~r − λ ·
Z
†
Ψ (~r)Ψ(~r)d~r − 1
By using the Lagrange multiplier λ, we have converted the optimization of E over
normalizable Ψ to an optimization of W over all Ψ and λ. For a given value of λ, we
extremize W over Ψ using the Euler-Lagrange equations
∂W
= HΨ(~r) − λ · Ψ(~r) = 0
∂Ψ(~r)
(1.2)
By taking the inner product of equation 1.2 against Ψ† , we find that λ = E. We
thus derive the Schroedinger equation:
HΨi (~r) = Ei Ψi (~r)
(1.3)
Equation 1.3 is a linear differential equation, and its solutions Ψi are the eigen20
states of the Hamiltonian H. The eigenstate with the lowest Ei is the ground state.
The Ψi form a complete orthonormal basis, and any state Φ can be expanded as a linP
ear combination of the Ψi , Φ(~r) = i Ci Ψi (~r), and the time-dependent Schroedinger
equation
HΦ(~r, t) = i
dΦ(~r, t)
dt
then gives its time-evolution as
Φ(~r, t) =
X
Ci Ψi (~r)e−Ei t
i
1.1.1
Hartree-Fock
For electrons, the wavefunction Ψ has to be normalized and completely antisymmetric. Both requirements can be satisfied by the Slater determinant form [6] for the
wavefunction. Hartree-Fock is based on this functional form, and is a mean-field theory, where each electron only feels the average effects of the other electrons on itself.
This simplifies the interaction into an effective one-body potential, which in turn allows the wavefunction of 3N dimensions to be expressed in terms of N 3-dimensional
spin-orbitals.
The Slater determinant takes the set of orthonormal spin-orbitals φi(x), where
x includes both spatial r and spin coordinates s, and constructs Ψ by taking the
determinant of matrix φi (xj ) where i and j are the row and column indices start from
1 and run up to the number of electrons N:
φ1 (x1 ) φ1 (x2 )
1 φ2 (x1 ) φ2 (x2 )
Ψ(x1 , x2 , · · · , xN ) = √ .
..
N! ..
.
φN (x1 ) φN (x2 )
Since the orbitals are normalized, the factor of
tion for the many-body wavefunction.
21
√1
N!
···
···
..
.
···
φ1 (xN ) φ2 (xN ) .. . φN (xN )
provides the right normaliza-
Since all electron positions are equivalent, the expectation of a one-body operator
in terms of the orbitals is then
Z
Ψ† (x1 , x2 , · · · , xN )
= N
=
Z
XZ
X
i
Oi Ψ(x1 , x2 , · · · , xN )dx1 dx2 · · · dxN
Ψ† (x1 , x2 , · · · , xN )O1 Ψ(x1 , x2 , · · · , xN )dx1 dx2 · · · dxN
φ†j (x1 )O1 φj (x1 )dx1
j
and the expectation of a two-body operator is
Z
Ψ† (x1 , x2 , · · · , xN )
X
i<j
Oij Ψ(x1 , x2 , · · · , xN )dx1 dx2 · · · dxN
N(N − 1)
Ψ† (x1 , x2 , · · · , xN )O12 Ψ(x1 , x2 , · · · , xN )dx1 dx2 · · · dxN
2
Z
1X
φ†j (x1 )φ†k (x2 )O12 φj (x1 )φk (x2 )dx1 dx2
(1.4)
=
2 j,k
Z
1X
φ†j (x1 )φ†k (x2 )O12 φj (x2 )φk (x1 )dx1 dx2
(1.5)
−
2 j,k
=
Z
The detailed derivation of these rules can be found in reference [7]. For the
inter-electron Coulomb operator vee , the integral corresponding to line 1.4 is simply
the classical Coulomb repulsion denoted EJ and is strictly positive, while line 1.5
corresponds to the energy savings that come from the anti-symmetry of the singledeterminant wavefunction, has a strictly negative contribution, and is called the exchange term, denoted as EK . The Hartree-Fock energy is then
22
EHF = ET + Evext + EJ − EK
XZ †
ET =
φi (r1)T1 φi (x1 )dx1
i
Evext =
EJ
EK
XZ
φ†i (r1)vext (x1 )φi (x1 )dx1
i
Z
1X
1
=
φ†j (x1 )φ†k (x2 )
φj (x1 )φk (x2 )dx1 dx2
2 j,k
|r1 − r2 |
Z
1
1X
φ†j (x1 )φ†k (x2 )
φj (x2 )φk (x1 )dx1 dx2
=
2 j,k
|r1 − r2 |
(1.6)
(1.7)
By minimizing EHF with respect to the orbitals, and enforcing normality of the
orbitals with Lagrange multipliers ǫi , one gets the Hartree-Fock equations
HHF φi = ǫi φi
(1.8)
(1.9)
where the effective single-particle Hamiltonian, the Fock operator HHF , is given by
HHF = T + vext + J − K
(1.10)
and the Coulomb J and exchange K operators have been defined as
!
X Z φ†j (x2 )φj (x2 )
Jf (x1 ) =
dx2 f (x1 )
|r1 − r2 |
j
!
X Z φ†j (x2 )f(x2 )
Kf (x1 ) =
dx2 φj (x1 )
|r
−
r
|
1
2
j
The eigenstates of the Fock operator are the Hartree-Fock orbitals. In this picture,
the electrons interact with each other purely through the Coulomb term J and the
23
exchange term K.
The Hamiltonian has no spin dependence, and hence commutes with the spin
operator. This makes the spatial and spin dependence of each spin-orbital separable
- each spin-orbital φi (x) can be expressed as φi (r)σi (s) where σ has two possible
values, α and β. Accounting for this explicitly, we get the unrestricted Hartree-Fock
equations
HU HF φi = ǫi φi
(1.11)
HU HF = T + vext + J − K
!
X Z φ†j (r2 )φj (r2 )
Jf (r1 ) =
dr2 f (r1 )
|r
−
r
|
1
2
j
!
X Z φ†j (r2 )f(r2 )
Kf (r1 ) =
dr2 φj (r1 )δσj ,σf
|r
−
r
|
1
2
j
where all the operators are now spatial integrals. The orbitals are occupied as in
figure 1-1.
So far, by assuming the single determinant form, we have reduced the search for
the N-electron wavefunction Ψ(x1 , x2 , · · · , xN ) to a search for N orbitals φi (r). Each
orbital at this point could be any 3-d function. We introduce a further simplification
of a basis set at this point. A basis set is a set of functions which is used to expand
the orbital functions: φi is represented as a linear combination of basis functions:
P
φi (r) = j Cij χj (r). Within a given basis set, the orbital coefficients Cij then specify
the orbital.
We now introduce a change in the representation of state. The one-body reduced
P
density matrix ρσ (r, r′ ) = i δ(σ, σi )φi (r)φ†i (r′ ) is a one-body projection operator onto
the occupied orbital space, and can be built from the orbitals. The UHF orbitals
can also be obtained from it (up to a unitary transformation) by diagonalization.
This operator contains the same information as the orbitals, and is generally more
convenient to work with.
Equation 1.11 is an integro-differential equation because of the J and K terms,
24
Figure 1-1: UHF orbital filling - ↑ are α-electrons, ↓ are β-electrons. Each horizontal
line represents a different orbital, and the α and β electrons fill different orbitals in
an Aufbau manner, from the bottom up. In this particular example, there are 4 α
electrons, 4 β electrons and 6 basis functions forming 12 spin-orbitals
and is usually solved using iterative techniques (fig. 1-2) which compute J and K
from {φi} and use numerical linear algebra to solve the eigenproblem in equation 1.11.
For the solution to equation 1.11 to be a minimum, the self-consistency criterion must
be satisfied, i.e. φi must be the eigensolution to the mean-field Fock matrix HU HF
computed from itself, which also means that the reduced density matrix ρσ (r, r′ )
commutes with the Fock matrix HU HF
Z
ρσ (r, r′ )HUHF(r′ , r′′ ) − HUHF (r, r′ )ρσ (r′ , r′′ )dr′ = 0
The Direct Inversion in Iterative Space (DIIS) step in figure 1-2 is an extrapolation
method that uses past Fock matrices to produce a better guess for the next iteration
[8, 9].
The Hartree-Fock energy is variational, i.e. it is always higher than the correct
ground-state energy.
25
Figure 1-2: Typical Algorithm for Solving the HF equations
EHF ≥ Eground
This is because the only approximation we have made is to restrict the search to the
space of single-determinant wavefunctions made of orbitals in some finite basis, which
is a subset of the space of all antisymmetric wavefunctions.
1.1.2
Electronic Correlation
Eigenstates of the full Hamiltonian (equation 1.1) are 3N-dimensional antisymmetric,
normalized functions. N-body eigenstates of non-interacting Hamiltonians are Slater
determinants, which are a subset of these functions. Under Hartree-Fock, all the
interactions come in the form of a effective one-body operator, the Fock operator,
which means many many-body effects are being neglected. This is why we call Slater
determinants non-interacting states.
The correlation energy is defined as the difference between the ground-state energy
and the Hartree-Fock energy in the complete basis limit, and is negative because of
26
the variational nature of the Hartree-Fock energy. Even in the complete basis limit,
the correlation energy is non-zero because the exact ground-state wavefunction is not
a single determinant.
Ec = Eground − EHF
As described before, the Slater determinant takes N spin-orbitals and constructs
an N-body wavefunction out of them. The non-interacting ground-state is the Slater
determinant constructed from the N lowest eigenvalue spin-orbitals. Non-interacting
excited-states are also Slater determinants, and are constructed from all the other
sets of N spin-orbitals. Altogether, from a basis of M spin-orbitals, M
Slater
N
determinants including the ground state can be formed. These are all the eigenstates
of a Hermitian N-body operator, and thus form a complete basis for expressing any
other N-body wavefunction, including the interacting ground-state Φ0 . Following
chapter 4 from Ref. [7], we denote the non-interacting ground state with Ψ0 and
denote the excited states by listing the orbital substitutions. For example, the excitedstate determinant that substitutes unoccupied orbitals r and s for occupied orbitals
a and b would be denoted as Ψrs
ab .
Φ0 = c0 Ψ0 +
X
ar
cra Ψra +
X
a<b
r<s
rs
crs
ab Ψab +
X
rst
crst
abc Ψabc +
a<b<c
r<s<t
X
a<b<c<d
r<s<t<u
rstu
crstu
abcd Ψabcd + · · · (1.12)
This is the CI expansion for the exact wavefunction. Given a basis of reference
spin-orbitals, usually from Hartree-Fock, the exact ground state is expressable in
terms of the coefficients c in equation 1.12. There are, however, M
coefficients, a
N
number which grows exponentially with the basis size - as of 2008, the biggest systems
treatable with Full CI are only of 10-12 electrons.
All differences between the exact wavefunction Φ0 and the Hartree-Fock wavefunction Ψ0 are bundled under the title of correlation. Correlation is further qualitatively
divided into two types, dynamical and static correlation.
27
Static correlation is the long-range correlation that results from there not being a
good single-determinant representation for the ground state. For example, the exact
electronic wavefunction in a dissociated singlet H· -H· molecule is equal parts H↑ -H↓
and H↓ -H↑ . However, in the large R limit, the best single-determinant wavefunction
is
1
1 √2 (φL↑ (r1 ) + φR↑ (r1 ))
Ψ(r1 , r2 ) = √ 2 √1 (φL↓ (r1 ) + φR↓ (r1 ))
2
√1 (φL↑ (r2 )
2
1
√ (φL↓ (r2 )
2
+ φR↑ (r2 ))
+ φR↓ (r2 ))
∝ {(φL↑ (r1 ) + φR↑ (r1 ))(φL↓ (r2 ) + φR↓ (r2 ))} − {r1 ↔ r2 }
= {φL↑ (r1 )φL↓ (r2 ) + φR↑ (r1 )φR↓ (r2 ) + φL↑ (r1 )φR↓ (r2 ) + φR↑ (r1 )φL↓ (r2 )}
− {r1 ↔ r2 }
The first two terms are energetically unfavorable ionic H−↑↓ -H+ and H+ -H−↑↓
terms which have energies that go as −1/R [10]. This is a case we discuss further in
chapter 3. Static correlation can be treated by explicitly using multiple determinants
in multi-reference methods [11].
Dynamic correlation is a short-range effect that results from electrons dynamically
avoiding each other and thus lowering inter-electron repulsion. A simple example
would be the Helium atom, which has two electrons in close proximity. It can be
treated starting from a single-determinant reference and subsequently adding contributions from excited determinants either variationally or perturbatively. In cases with
dynamic correlation, the actual wavefunction may have a good overlap (90%-95%)
with the single-determinant wavefunction, with the remaining bit being the result of
summing small contributions from many other determinants. Those small contributions are significant energetically because they allow the electrons to move in such
a way as to avoid each other. These contributions are best treated by perturbative
methods like MP2 [12].
28
Active Space Methods
Active space methods solve the Schroedinger equation within a subspace of the
M
N
determinant complete basis. That subspace is known as the active space, and CI
methods use the matrix elements of the full electronic Hamiltonian, and the overlap
within that space to compute the c coefficients. The relatively small number of
determinants needed to treat static correlation motivates the use of active space
methods.
For example, in Complete-Active-Space Self-Consistent-Field (CASSCF), one picks
a subset of the M orbitals Ms and number of electrons Ns , and within the resulting
Ms
space of determinants, alternately solves for the CI coefficients c and the optimal
Ns
orbitals {φi } under a given set of CI coefficients.
Figure 1-3: Some CASSCF(2,3) configurations
In figure 1-3 we have shown some possible configurations under CASSCF(2,3),
2
where 2 electrons are placed in any of the 31 possible configurations with the same
total spin.
We develop a DFT-based active space CI method for treating both dynamic and
static correlation in chapter 3.
29
1.2
1.2.1
Density Functional Theory
The Hohenberg-Kohn Theorems
Most of the difficulties in conventional wavefunction-based electronic structure stem
from the complexity of the many-body wavefunction Ψ(~r). It is, however, possible to
avoid the many-body wavefunction. Density Functional Theory (DFT) is the study
of electronic structure with single-particle density as the independent parameter. The
single-particle density is obtained from the many-body wavefunction by integrating
out all but one position variable.
ρ(x1 ) =
Z
Ψ† (x1 , x2 , · · · , xN )Ψ(x1 , x2 , · · · , xN )dx2 dx3 · · · dxN
The single-particle density is a vastly simpler object than the many-body wavefunction, having only 3, as opposed to 3N, coordinates. Surprisingly, as proven by the
Hohenberg-Kohn theorem [13], the single-particle density contains all the information
of the many-body ground-state wavefunction. The crux of the Hohenberg-Kohn theorem is the idea that, even though electronic Hamiltonians are two-body operators,
the difference between any two electronic Hamiltonians is a one-body operator.
We will now present a proof via reductio ad absurdum for the first HohenbergKohn theorem, which states that there exists a one-to-one correspondence between
densities ρ and external potentials vext . We start by assuming the converse, that there
exist two external potentials vext1 and vext2 which differ by more than a constant
which give rise to two different wavefunctions Ψ1 and Ψ2 and two different energies
E1 and E2 but the same density ρ. The expectation of E1 − E2 is given by
30
E1 − E2 =
=
=
Z
Z
Z
Ψ†1 (~r)H1 Ψ1 (~r)d~r
−
Ψ†1 (~r)H1 Ψ1 (~r)d~r
−
Ψ†1 (~r)H1 Ψ1 (~r)d~r
−
Z
Z
Z
Z
Ψ†2 (~r)H2 Ψ2 (~r)d~r
Ψ†2 (~r)(H2 − H1 + H1 )Ψ2 (~r)d~r (1.13)
Ψ†2 (~r)H1 Ψ2 (~r)d~r
Ψ†2 (~r)(H2 − H1 )Ψ2 (~r)d~r
Z
Z
†
=
Ψ1 (~r)H1 Ψ1 (~r)d~r − Ψ†2 (~r)H1 Ψ2 (~r)d~r
Z
−
ρ(~r)(vext2 (~r) − vext1 (~r))d~r
−
(1.14)
By the variational principle, we know that
Z
Ψ†1 (~r)H1 Ψ1 (~r)d~r
−
Z
Ψ†2 (~r)H1 Ψ2 (~r)d~r < 0
(1.15)
and so
E1 − E2 < −
Z
ρ(~r)(vext2 (~r) − vext1 (~r))d~r
(1.16)
We could, however, also have chosen to do the same proof for the expectation of
E1 − E2 , which would merely have swapped the labels 1 and 2, giving us
E1 − E2 >
Z
ρ(~r)(vext1 (~r) − vext2 (~r))d~r
(1.17)
Combining inequalities 1.16 and 1.17, we get the contradiction
E1 − E2 > E1 − E2
Thus it must not be possible for two external potentials differing by more than a
constant to give rise to the same density. This proves the existence of a mapping from
densities ρ(~r) to external potentials vext (~r). The converse mapping is simply solving
the Schroedinger equation for external potential vext (~r) and integrating the resulting
31
wavefunction to obtain the density ρ(~r). This is the first Hohenberg-Kohn theorem
- the existence of a one-to-one map between the set of single-particle ground-state
densities and the external potentials which generate them.
Since the external potential defines the ground-state wavefunction from which
every ground-state observable can be obtained, the first Hohenberg-Kohn proof establishes the existence of density functionals for every property of the corresponding
ground-state wavefunction. This includes, crucially, its energy expectation under any
other Hamiltonian. Using this fact, the second Hohenberg-Kohn theorem expresses
the variational principle in a density functional form, proving the existence of a universal functional G[ρ] which can be used to find the energy of any ground-state density
under any other Hamiltonian as defined by external potential vext (r)
Ev [ρ] =
Z
1
vext (r)ρ(r)dr +
2
Z
ρ(r)ρ(r′ )
drdr′ + G[ρ]
|r − r′ |
(1.18)
where G[ρ] is universal and only dependent on the density, and Ev [ρ] is variational.
Even though it is proven to exist, however, the explicit form of the energy functional
G[ρ] is not known. There are, instead, numerous attempts at approximations to the
energy functional. One of the key approaches, in which the work of this thesis is
couched, is the Kohn-Sham formalism [14].
1.2.2
The Kohn-Sham Formalism
To derive the Kohn-Sham formalism, one begins by considering the set of noninteracting Hamiltonians
Hs = −
1 2
∇ + vext
2
(1.19)
Notice that the Hohenberg-Kohn theorem also applies for this set of non-interacting
Hamiltonians Hs, and therefore a one-to-one mapping exists between the non-interacting
ground-state densities ρ and vext which give rise to them. A density functional of the
non-interacting kinetic energy, Ts [ρ] then exists. All the ground states to Hamiltoni32
ans 1.19 are single determinants, and Ts can be computed exactly from the orbitals
that make up the determinant
ρ(r) =
XZ
φ†i (r)φi (r)
i
Ts [ρ] =
XZ
φ†i (r)
i
1 2
∇ φi (r)dr
2
Using Ts , Kohn and Sham define Exc , the exchange-correlation energy, as such:
G[ρ] = Ts [ρ] + Exc [ρ]
(1.20)
Substituting G[ρ] into equation 1.18 then gives
Ev [ρ] = Ts [ρ] +
Z
1
vext (r)ρ(r)dr +
2
Z
ρ(r)ρ(r′ )
drdr′ + Exc [ρ]
|r − r′ |
(1.21)
Extremizing Ev [ρ] with respect to Kohn-Sham orbitals φi (r) and using the EulerLagrange equations then gives the Kohn-Sham equations
HKS φi (r) = ǫi φi (r)
dEv [ρ]
HKS =
dφi (r)
1
= − ∇2 + vext + J + vxc [ρ]
2
dExc [ρ]
vxc [ρ] =
dρ
Kohn-Sham DFT is a search over N-body single determinant space, and is solved
using highly similar techniques to Hartree-Fock, like that depicted in fig. 1-2. The
Kohn-Sham wavefunction is the exact ground-state solution to a non-interacting
Hamiltonian, the Kohn-Sham Hamiltonian. The resulting density is assumed to also
be a fully-interacting ground-state density corresponding to some other external po33
tential vext (r). This is the assumption of v-representability. This is a requirement
inherited from the Hohenberg-Kohn theorem, which is the proof of a one-to-one mapping only within a subset of all possible ground-state densities - densities which originate from the ground state under some vext . Examples of densities which are not
v-representable include one-electron excited state densities, and also other specific
examples invented to explore this assumption [15, 16]. In the absence of a rigorous
proof, however, KS-DFT has been applied to many ground-state molecular systems
and found to work quite well empirically. This suggests that non v-representable
densities must be quite rare in practice. We explore this issue further in chapter 4.
Given v-representability, Kohn-Sham DFT (KS-DFT) is in principle an exact
formulation of many-body quantum mechanics. However, just like the case for the
energy functional Ev [ρ], the expression for Exc [ρ], the exchange-correlation functional,
is not known. The achievement of KS-DFT lies in the non-interacting kinetic energy
Ts [ρ] being a good approximation to the exact kinetic energy, making Exc [ρ] much
smaller in magnitude than G[ρ], and correspondingly easier to approximate. Despite
this, the development of exchange-correlation functionals remains an active field of
research, and the quality of the exchange-correlation functional used has a big effect
on the efficacy of KS-DFT.
1.2.3
Constrained Density Functional Theory (CDFT)
KS-DFT searches for the lowest energy solution amongst all possible single-determinant
Kohn-Sham wavefunctions. Constrained DFT is a search that operates only over the
space of Kohn-Sham wavefunctions that satisfy the specified density constraints [17].
Constraints come in the form of
Z
wc (r)ρ(r)dr − Nc = 0
(1.22)
where wc is a space dependent weight function, ρ(r) is the electron density and Nc
is the target value of the constraint operator. Spin indices have been suppressed for
clarity. Possible constraints are any function of position, i.e. dipole and multipole
moments and the total electron density on a subset of the atoms. Some operators like
34
the momentum operator cannot be expressed in this form.Constraints are imposed
using Lagrange multipliers, such that instead of finding the minimum of Ev [ρ] we find
the extrema of W [ρ, Vc ]:
W [ρ, Vc ] = Ev [ρ] + Vc
Z
wc (r)ρ(r)dr − Nc
We only show one constraint here, but the extension to multiple constraints is
straightforward. The resulting Kohn-Sham equations then look like they have an
extra external potential Vc wc (r) added. This potential is a function of the density,
and CDFT introduces no new assumptions over those already needed for KS-DFT.
δW [ρ, Vc ]
= HCDF T = HKS +Vc wc
δρ
HCDF T φi (r) = ǫi φi (r)
(1.23)
(1.24)
The derivative with respect to Vc must also be zero when the constraint is satisfied:
* 0 ∂W [ρ, V ]
dW [ρ, Vc ]
dρ δW [ρ,V
c
c]
=
+
dVc
dV
δρ
∂Vc
Z c
=
wc (r)ρ(r)dr − Nc = 0
(1.25)
Note that we have made use of the fact that δW [ρ, Vc ]/δρ = 0 (Equation 1.23). Second
derivatives come from first-order perturbation theory:
d2 W [ρ, Vc ]
=
dVc2
dρ(r)
dr
dVc
R
X X | φi (r)wc (r)φa (r)dr|2
= 2
ǫi − ǫa
i∈occ a∈virt
Z
wc (r)
(1.26)
where indices i and a run over the occupied and unoccupied orbitals respectively, and
ǫi,a are the Kohn-Sham eigenvalues. Equation 1.23 is solved as an eigenproblem using
35
numerical linear algebra routines, while equation 1.25 is solved using root-searching
algorithms, with the help of the analytic second derivative equation 1.26.
Figure 1-4: Algorithm for Solving the CDFT equations
Figure 1-4 illustrates the algorithms we use to solve the CDFT equations. within
each of the black boxes are additional iterations which add the right multiple of
the constraint matrix to the output Fock matrix, set such that the resulting density
matrix built from the Fock matrix eigenvectors satisfies the constraints in equation
1.25.
36
Possible Constraints
Formally, the constraint wc can be any function of the α- and β- densities, but in
our implementation is limited to linear combinations of α- and β- atomic populations.
We have implemented two definitions for the assignment of atomic populations, Becke
weights and Lowdin populations. The Becke weights use the atomic center assignments of density that is used in the Becke quadrature [18], whereas Lowdin weights
are basis-dependent projection operators obtained from diagonalizing the overlap matrix in the AO basis and assigning the resulting orthogonal basis functions to the AO
basis atomic centers. In either case, the resulting populations sum to the total density
of the atom and the population operators are Hermitian.
Previous work in the group has applied CDFT to the computation of long-range
charge-transfer states [19], electron transfer coupling coefficients and reorganization
energies [20, 21], all through the use of different constraints. We will use CDFT in
chapter 3 to construct diabatic states, compute their electronic couplings, and use
that information to construct a more accurate ground state.
1.2.4
Time-Dependent Density Functional Theory (TDDFT)
So far, we have only talked about ground-state densities. There is, however, also
a formulation of time-dependent quantum mechanics based on the single-particle
density. As described earlier, the basis of the Hohenberg-Kohn theorem is the fact that
differences within the space of fully-interacting electronic Hamiltonians are all oneparticle operators. Analogously, within the space of time-dependent fully-interacting
electronic Hamiltonians, if we disallow magnetic fields, any difference is also a timedependent one-particle potential. For time-dependent Hamiltonians of the form
H(t) = T + vext (t) + vee
the Runge-Gross theorem [22] states that, given initial ground-state wavefunction
Φ(0), there exists a one-to-one correspondence between the time-dependent potential
vext (t) and the time-dependent density ρ(t). The corresponding Kohn-Sham formalism to the Runge-Gross theorem is the time-dependent Kohn-Sham (TDKS) equation,
37
which expresses the evolution of Kohn-Sham orbitals in response to a time-dependent
external potential:
∂t φi (t) = −i · HKS[{ρ(τ )}τ <t , t]φi (t)
(1.27)
Just like the case for the ground-state Kohn-Sham equations, the time-dependent
Kohn-Sham orbitals give the same density as the exact wavefunction, but otherwise
are not to be attributed any physical meaning. Notice the density dependence in
equation 1.27 is of all densities before time t.
Time-Dependent Density Functional Response Theory (TDDFRT)
TDDFT is usually used in its linear response form, linear TDDFRT. We follow the
treatment presented in Ref. [23]. Consider, to first order, the effect of a perturbation
of time-dependent external potential at time t, δvext (t) on the TDKS density matrix
at time t′ , ρ(t′ ) = ρ0 + δρ(t′ ). The system starts out at t = −∞ in the ground state
ρ0 , is subject to perturbation δvext (t), and obeys the TDKS equations
1 2
∇ + vext + δvext (t) + J[ρ] + vxc [ρ]
2
∂t ρ(t) = −i[HKS [ρ(t), t], ρ(t)]
HKS [ρ, t] = −
(1.28)
The response of a system at time t′ to a perturbation at time t is purely a function
of τ = t′ − t and of the density at t, ρ(t). This time-translation invariance makes
it preferable to work in frequency space, so we assume the following monochromatic
forms for δvext (t) and δρ(t′ ).
†
δvext (t) = δṽext (ω)e−iωt + δṽext
(ω)eiωt
δρ(t′ ) = δρ̃(ω)e−iωt + δρ̃† (ω)eiωt
38
To first order, the TDKS equation of motion (Eq. 1.28) then results in



 



 A(ω) B(ω)
1 0  δρ̃(ω)
δṽext (ω)

−ω
 
 = 

†
 B(ω) A(ω)

†
0 −1
δρ̃ (ω)
δṽext (ω)
Aiajb (ω) = (ǫa − ǫi )δij δab + Biabj (ω)
Z
1
′
+ f̃xc (ω, x, x ) φb (x′ )φj (x′ )dxdx′
Biajb (ω) =
φi(x)φa (x)
r − r′
(1.29)
(1.30)
where φi are the Kohn-Sham orbitals from the ground-state and ǫi are the Kohn-Sham
eigenvalues. f˜xc (ω, x, x′ ) is the frequency-dependent exchange-correlation kernel, and
represents the change in vxc at one time in response to a change in the density at
another. Up until this point, the treatment has been exact, given the restriction of
sticking to purely electric fields. Note that this means we are unable to treat magnetic
phenomena like EPR or electronic circular dichroism.
Under the adiabatic approximation, vxc is assumed to depend only on the density
at the same time, which in frequency space translates to f̃xc having no frequency
dependence:
f̃xc (ω, x, x′ ) = f̃xc (0, x, x′ ) =
δvxc [ρ](x)
δ 2 Exc [ρ]
=
δρ(x′ )
δρ(x)δρ(x′ )
(1.31)
When the LHS of equation 1.29 is zero






A(ω) B(ω)
δρ̃(ω)
1 0
δρ̃(ω)


 = ω

,
†
†
B(ω) A(ω)
δρ̃ (ω)
0 −1
δρ̃ (ω)
a finite δṽext (ω) gives rise to an infinite δρ̃(ω) - these are the poles of the response
matrix
−1

A(ω) B(ω)


B(ω) A(ω)
39
,
and the corresponding ω are the excited state energies of the system, with δρ̃(ω) being
the Kohn-Sham transition density matrices, which in turn give the exact transition
densities.
1.2.5
Approximate Exchange-Correlation Potentials
As mentioned earlier, the exact form for Exc [ρ] is not known, even in principle.
In order to use KS-DFT, we must have a way of approximating Exc [ρ]. Unlike
wavefunction-based approaches, there is no exact space which is truncated to produce
approximations - functionals are picked to satisfy both computability constraints and
also to satisfy criteria that the exact functional is known to satisfy, and maybe also
to produce the correct result for a model system. In this subsection, we introduce the
exchange-correlation functionals which we use in our work.
The Local Spin Density Approximation (LSDA) The uniform electron gas
is a fictitious system where a negative sea of electrons exactly neutralizes a uniform
background of positive charge. It is parameterized by only two parameters, the alpha
density ρα and the beta density ρβ , and so, assuming the form
Exc [ρα , ρβ ] =
Z
LSDA
fxc
(ρα (r), ρβ (r))dr
LSDA
it is possible to formulate functional fxc
(ρα , ρβ ) which gives the exact exchange and
correlation energies for every possible uniform electron gas. The resulting functionals
are the LSDA exchange and correlation functionals. The exchange functional was
derived by Slater as an approximation to Hartree-Fock exchange [2], whereas the
correlation functional is an interpolation based on accurate Monte-Carlo simulations
[24].
The LSDA works very well in situations where the electron density varies slowly,
outperforming semi-empirical functionals for solids and jellium surfaces, but tends to
overbind electrons in molecules and atoms [25].
Generalized Gradient Approximations (GGAs) One of the earliest attempts
to go beyond the LSDA was the gradient expansion approximation (GEA), a trun40
cated expansion based on slowly varying electron gas densities. This functional was
found to have worse performance than the LSDA when used on molecules.
This was found to be due to a violation of certain exact conditions called the
exchange-correlation hole sum rules [26, 27, 28]. By modifying the GEA functional to
satisfy the exchange-correlation hole sum rules, the first GGAs like PW86 [27] were
formulated. GGAs are of the form
Exc [ρα , ρβ ] =
Z
fxc (ρα (r), ρβ (r), |∇ρα (r)|, |∇ρβ (r)|)dr
and make use of both density and gradient to determine functional values. Later
GGAs like PW91 [29] and PBE [30] were derived by first coming up with a model
for the exchange-correlation hole, and then deriving the resulting functional. The
judicious selection of exact conditions to satisfy is crucial - of particular note is the
Becke88 [31] exchange GGA functional, which gives the right asymptotic exchange
potential for finite systems, on top of satisfying the exchange hole sum rules.
GGAs correct for the tendency of LSDA to overbind and overestimate bond energies, performing better than LSDA in predicting total energies, atomization, barriers
and geometries [32, 33, 34, 35].
The Adiabatic Connection and Hybrid Functionals Consider the set of Hamiltonians
λ
Hλ = T + vext
+ λ · vee
The Kohn-Sham and the full Hamiltonians are instances of this set, with λ = 0
λ
and λ = 1 respectively. For a given density ρ(r), assume that there exists a vext
for every value of λ that gives rise to ρ(r) as its ground-state density, with corresponding wavefunctions Ψλ . Then, using the Hellman-Feynman theorem, one finds
an expression for the exchange-correlation energy Exc [ρ] [36]:
Exc [ρ] + EJ [ρ] =
Z
1
0
41
Ψ†λ (~r)vee Ψλ (~r)d~r
(1.32)
This process is known as the adiabatic connection. At λ = 0 , the integrand
of equation 1.32 is ∝ J − K, and the wavefunction Ψ0 is the Kohn-Sham Slater
determinant. This implies that if we were to evaluate integral 1.32 by quadrature, the
sum would include some component from K evaluated on the Kohn-Sham orbitals.
This logic guided Becke to postulate the half-and-half functional in Ref. [37] and
the B3PW91 functional in Ref. [38]. These are the hybrid functionals, some linear
combination of GGAs and the Hartree-Fock exchange as evaluated using the KohnSham orbitals. In the case of B3PW91, the coefficients of the linear combination
were determined empirically by fitting to experimental results [38]. The resulting
functional is
B3PW91
LSDA
Exc
= Exc
+ a0 (Exexact − ExLSDA ) + ax (ExBecke88 − ExLSDA ) + ac (EcPW91 − EcLSDA )
where a0 = 0.20, ax = 0.72, ac = 0.81.
Currently, the most popular hybrid functional is B3LYP, which uses the same
coefficients as B3PW91 but replaces the PW91 correlation functional with the LYP
correlation functional. By incorporating exact exchange, B3LYP is able to get an
accuracy of 2-3 kcal/mol within the G2 test set. This is very close to chemical
accuracy, ≈ 1 kcal/mol. Also because of the exact exchange component, hybrid
functionals exhibit a lower level of self-interaction error (SIE) proportional to the
amount of exact exchange [39]. We discuss SIE further below.
One might think to just use exact exchange and only approximate correlation.
The problem with this approach lies in the inability of a delocalized exchange hole, in
combination with a localized correlation hole, to give rise to a good approximation to
the exact exchange-correlation hole (Ref. [10] chapter 6.6). Hybrid functionals hence
represent a compromise between accurate exchange and accurate correlation.
Advantages and Flaws of Approximate Exchange-Correlation Functionals
The choice to use DFT is often motivated by its efficiency. DFT is as cheap as HartreeFock theory for most purposes because of the way commonly used xc potentials
42
are designed - Exc [ρ] are formulated such that the resulting vxc [ρ] =
dExc [ρ]
dρ
can
be evaluated point-by-point on a spatial grid. The consequence of this is that the
computation required to calculate vxc scales linearly with system size. This is what
makes KS-DFT so efficient.
KS-DFT’s efficiency in treating electronic correlation allows it to be used on systems for which conventional wavefunction methods are simply too expensive. This
cheapness, however, comes at a price - we know of many characteristics of the exact xc
functional which that commonly used approximations do not satisfy. The inaccuracies
are discussed below.
Lack of Derivative Discontinuity We first define the energy of densities that do
not integrate to integers as being the result of ensembles of integer densities. In Ref.
[40], Perdew et al. show that the lowest energy mixture for a density that integrates
to M + ω electrons, where M is an integer and 0 ≤ ω ≤ 1, is the mixture of (1 − ω)
of the lowest energy M-electron density and ω of the (M + 1)-electron density, giving
rise to the energy
EM +ω = (1 − ω)EM + ωEM +1
This means that the energy as a function of N is a series of straight lines with
discontinuities on the first derivative at integer values. As a consequence of this, the
Kohn-Sham HOMO eigenvalue is in fact discontinuous at integer N:
ǫHOM O =


−IZ

−AZ
(Z − 1 < N < Z)
(Z < N < Z + 1)
where IZ and AZ are the ionization and electron affinities of the Z-electron system. This property is not present in approximate non-hybrid exchange-correlation
functionals [40, 41, 42]. It is partly present in hybrid functionals because of the
orbital-dependence of exact exchange. The derivative discontinuity problem is most
evident in the separation of ionic diatomics, like LiH, which dissociates to erroneous
partial charges. Under LSDA, LiH dissociates to Li+0.25 H−0.25 instead of the correct
43
Li0 H0 [40]. In solid state applications, the lack of a derivative discontinuity causes
underestimates in band gap predictions, which at times makes semi-conductors and
insulators look like metals [43]. For example, it causes LSDA to predict a silicon band
gap that is 50% too small [44].
Spatial Locality The local nature of exchange-correlation approximations means
that charge transfer and charge transport cannot be treated correctly using perturbation theory [45].
As noted earlier, other than the exact exchange component of hybrid functionals,
common approximate exchange-correlation potentials can all be evaluated on a grid.
This means that the TDDFRT exchange-correlation kernel is not only local in time
due to the adiabatic approximation, but also local in space. Equation 1.31 then
becomes
f̃xc (0, x, x′ ) = f̃xc (0, x)δ(x − x′ )
The potential at a given location vxc [ρ](r) only depends on properties of the
density at the same point. This property is purely an ad-hoc simplification, and not
given by the original Kohn-Sham derivation.
The simplification of spatial locality leads to erroneous predictions for chargetransfer under linear response TDDFT [45, 46]. Consider the Biajb (ω) term of TDDFRT
in equation 1.30. The ia and jb indices refer to excitations φi → φa and φj → φb .
Note that for long-range charge transfers, where there is little overlap between these
excitation pairs, the B(ω) term effectively goes to zero, and the excitation is then
determined solely by the A(ω) contribution, the difference in orbital eigenvalues.
For well-separated molecules, this difference is not distance dependent at all, and
so does not give rise to the correct
1
R
factor. Charge transfer excitations are hence
underestimated by TDDFRT. The exact exchange component of hybrid functionals
is non-local, adding a multiple of the term
44
1
φa (x′ )φb (x′ )dxdx′
= − φi (x)φj (x)
r − r′
Z
1
X
Biajb (ω) = − φi (x)φb (x)
φa (x′ )φj (x′ )dxdx′
′
r−r
Z
AX
iajb (ω)
to Aiajb (ω) and Biajb (ω). AX
iajb (ω) has the correct
1
R
dependence and allows hybrid
functionals to perform better, but as a whole still exhibit the same problem.
Self-Interaction Error (SIE) This is the tendency of individual electrons to feel
an erroneous repulsion from their own densities [47, 48]. Notice that even though
the inter-electron interaction is pairwise between the
N (N −1)
2
pairs of electrons, the
sums in Hartree-Fock Coulomb (equation 1.6) and exchange (equation 1.7) energies
run over all N 2 orbital indices. This transformation is allowed because the j = k
terms in the Hartree-Fock Coulomb and exchange terms are equal and opposite in
sign to each other. This is the transformation that allows the Coulomb energy to
be evaluated purely from the classical electron density. It is, however, a problem for
DFT because DFT exchange is not evaluated exactly. Electrons hence experience a
spurious repulsion from themselves.
-0.4
Hartree-Fock (exact)
LSDA
PW91
B3LYP
Energy/Hartrees
-0.45
-0.5
-0.55
-0.6
-0.65
0
0.5
1
1.5
2
2.5
3
Bond Length/Angstroms
3.5
Figure 1-5: H+
2 Energies in cc-PVTZ basis
45
4
Consider the example of one-electron system H+
2 . Hartree-Fock does not suffer at all
from SIE, and is used to find the exact energy in this example. The error in the energy
difference between the equilibrium and dissociated states comes from there being a
larger SIE in the equilibrium state, when the electron density is more compact. Note
that due to the inclusion of some exact exchange, hybrid functionals suffer less from
SIE.
Lack of Static Correlation Wavefunctions which are well-described by the sum of
multiple determinants exhibit long-range correlations which are treated with multiplereference methods under wavefunction theories [11]. The Kohn-Sham Slater wavefunction is a single determinant by definition, and in principle is able to represent the
multiple determinant wavefunction just fine. This non-interacting v-representability
has been demonstrated for the specific case of dissociating H2 , where the wavefunction
in the dissociated limit is the sum of two determinants, and yet can be represented
by a single determinant KS wavefunction [49]. Using approximate xc functionals,
however, static correlation is usually underestimated by DFT, especially when compensating effects from SIE are eliminated [50].
-0.75
-0.8
Energy/Hartrees
-0.85
-0.9
-0.95
-1
-1.05
CCSD (Exact)
Hartree-Fock
LSDA
PW91
B3LYP
-1.1
-1.15
-1.2
0
0.5
1
1.5
2
2.5
3
Bond Length/Angstroms
3.5
4
Figure 1-6: H2 Energies in cc-PVTZ basis
Consider the 2-electron example of H2 . In 2-electron systems, the coupled-cluster
singles-doubles (CCSD) method is exact. None of the other methods have static
46
correlation, and in the case of Hartree-Fock this results in charge-separation terms
H−↑↓ -H+ and H+ -H−↑↓ which cause the energy to go up with distance. Note that
PW91, a GGA, and B3LYP, a hybrid functional, get pretty good energies at the
equilibrium distance due to their ability to treat dynamic correlation.
There is much work on new xc functionals which address these systemic inaccuracies [51, 52, 53]. In this work, however, no new xc functionals will be presented.
Instead, we focus on developing methods to extract physical properties of interest
using the DFT formalism. We do not ignore the artifacts described above; instead
of trying to tackle them directly, we design our procedures so as to minimize their
effects.
1.3
Structure of this Dissertation
This dissertation will be structured as follows - chapters are arranged in accordance
to physical properties of interest. Each chapter begins with an introduction to the
significance of the property being studied, followed by a description of the state of the
art, then a description of the methodology which we have developed. An account of
results from applying the methodology to physical systems follows. The chapter then
concludes with rationalizations of the failures and successes of the approach and, if
appropriate, possible future directions.
Chapter 2 covers the work that we have done to simulate conductance in single
molecules. We numerically integrate the time-dependent Kohn-Sham (TDKS) equations in a Gaussian basis to simulate the flow of charge across a polyene molecule,
and characterize the corresponding current-voltage characteristics of the molecule.
Chapter 3 describes a configuration interaction method based on constrained density functional theory, the advantage of which is its ability to treat both dynamic
and static correlation at a modest computational cost. We apply this method to the
computation of molecular dissociation curves of small diatomic molecules. Chapter
4 details explorations into the ability of DFT to treat isolated excited states - we
devise a method that uses non-Aufbau occupation within the Kohn-Sham formalism
to determine the energy of excited stationary states. This method is applied to the
47
Rydberg states of hydrogen and lithium atoms under various functionals.
48
Chapter 2
Molecular Conductance using
Real-Time Time-Dependent
Density Functional Theory
Note: The bulk of this chapter has been published in Ref. [54].
2.1
Introduction
Molecular electronics represent the next length-scale in the miniaturization of electronics as embodied in Moore’s Law [55]. The pertinent physics in molecular electronics differs from that of molecules in bulk solids, and experiments probing the nature of
single-molecule devices involving metal-molecule-metal (MMM) junctions have been
performed in an attempt to understand more about them [56, 57, 58, 59, 60, 61, 62,
63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79]. In this chapter, we
describe the development of DFT-based methods used to simulate molecular-scale
conductors.
2.1.1
The Landauer-Büttiker Model
We start by examining charge transport under the Landauer-Büttiker model [80, 81,
82, 83, 84]. The Landauer formula expresses the conductance Γ as a sum of M
channels which each have transmittance Ti .
M
Γ=
1 X 2
|Ti |
2π i
49
The Landauer-Büttiker model is an elastic electron transport model - electrons
only exchange momentum with the channel, and are either transmitted through or
reflected from the channel without a change in energy. Under the Landauer-Büttiker
model, the metal leads to which the channel is connected exist in a state of grand
canonical equilibrium, possesing well-defined chemical potentials and driving electron transport through the channel by virtue of the difference in chemical potential
between the two leads. The resistance is quantized as a function of chemical potential difference, with step increases in conductance corresponding to the activation
of new quantum channels. When applying the Landauer-Büttiker model to MMM
experiments, treatments that interpret the channels as purely molecular scattering
coefficients have been pursued [85, 86, 87, 88, 89]. Under such interpretations, the
molecule acts as a potential off which electrons of the bulk scatter.
2.1.2
Non-Equilibrium Green’s Function (NEGF) Methods
Beyond scattering theory, the non-equilibrium Green’s function [90] can also be used
to obtain a Landauer-like expression for electron transport through junctions. The
NEGF contains the information needed to treat conductance exactly, but obtaining the exact NEGF is as difficult as solving the many-body Schroedinger equation.
Therefore, instead of computing the exact NEGF, various approximations to the
NEGF have been developed [91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104].
DFT-based NGEF models use ground-state DFT to provide the effective singleparticle potential for the scattering formalism. They divide the system into three
regions, the left lead L, right lead R and molecule M, with the corresponding division
of the DFT Fock matrix

FL
FLM
0





Ftot = F†LM FM FM R  ,


†
0
FM R FR
For any given energy level, an effective Hamiltonian over the molecular basis
functions is built:
50
Fef f = FM − F†LM [FL − EI]−1 FLM − F†M R [FL − EI]−1 FM R
Fef f is then used as an effective one-body Hamiltonian for computing the NEGF.
Since the ground-state Fock matrix is used, the assumption implicit in this procedure
is that the steady state is a small perturbation from the ground state of the system.
There remain persistent discrepancies between theoretical prediction and experimental measurement of molecular conductance. One finds that the best experiments
and theories differ by two orders of magnitude in predicting the conductance of simple
junctions.
2.1.3
Time-Dependent Density Functional Theory
TDDFT can be used to study molecular conductance by simulating the evolution of
the electron density ρ(t) under a time-dependent potential v(t). By formulating the
appropriate potential v(t), the charge transport observed can be taken to represent
the flow of charge in the molecule in response to a voltage bias. This approach is
supported by formal justification for TDDFT conductance simulations [105, 106, 107],
the study of the effect of existing DFT approximations on conductance [108, 109] and
protocols for simulating current flow using TDDFT [110, 111, 112, 113]. Our work is
most directly preceded by that described in Refs. [110, 112, 113].
As described in the introduction chapter, time-dependent density functional theory is an exact reformulation of many-body quantum mechanics. As such, barring
approximations at the DFT level, one would expect such an approach to be exact.
Difficulties with using TDDFT
There are two main obstacles that must be surmounted in order to use TDDFT to
conduct full numerical simulations of molecular conductance.
Computational Expense The first difficulty is that of overcoming the computational expense of simulating the systems in real time. The conductance of a molecule
in theory is a local molecular property that is independent of the leads, and simulations should in principle model molecules as open systems that lose and gain electrons
51
from their environment. In practice, however, large closed systems that include the
environment are often used instead. Using a closed system to represent an open one
places lower bounds on the size of the system used - the simulated system has to be
large enough for results to approach convergence with respect to the infinite-size limit
[105]. There is also the issue of temporal convergence - experimental timescales are
typically much longer than electronic timescales, and hence experimental measurements are of steady-state properties. Therefore, for simulation results to be comparable to experiment, they must allow time for the system to reach steady state. These
lower bounds on system size and simulation time impose considerable computational
expense on the simulateur.
Despite this expense, however, simulations of credible size (e.g. 50-100 atoms)
and time spans (femtoseconds) can now be done [114, 115, 116, 117, 118]. It has also
been previously shown that one can attain a quasi-steady state in a simple metal
wire using TDDFT [112]. Key to making these simulations practical is the efficient
integration of the time-dependent Kohn-Sham (TDKS) equations [119, 120, 121].
Voltage Identification The second major obstacle to TDDFT conductance simulations is the difficulty of assigning the voltage value to a given current simulation. This is not a problem within the Landauer formalism, since the leads are
always near equilibrium, the voltage is simply the difference between two well-defined
chemical potential values. Differences in effective single-particle energy levels are
used in some modern theories to define voltage, the choices being the difference between Fermi-levels conditioned on direction of propagation [122, 91, 123, 124, 125],
or Fermi-levels of regions deep within the leads, away from the conducting molecule
[94, 95, 96, 97, 98, 99, 100, 101]. The use of Fermi-levels in DFT is not fully justified,
as the idea of a Fermi level is only valid for non-interacting particles. KS orbital
energies (other than the HOMO), which are used as proxies for Fermi energies, are
known to be meaningless [40, 41, 42, 126], and as a result, when computing biases by
subtracting two KS orbital energies from each other, one is in fact using an uncontrolled approximation. This difficulty is specific to the DFT formalism, and different
in nature from those faced in identifying the experimental voltage with microscopic
52
reality.
2.2
2.2.1
Methodology
Integrating the TDKS equations
These are the TDKS equations:
HKS[ρ(t), t] · φi (t) = i
where ρ(t) ≡
Pocc
i
dφi (t)
dt
(2.1)
|φi (t)|2 . They describe the evolution of a density in response
to time-dependent external potential. Under the KS formalism, an effective noninteracting mean-field Hamiltonian HKS [ρ(t), t] is computed from the density ρ(t),
and it is under this Hamiltonian that the KS orbitals φi (t) evolve. Under the adiabatic
approximation [127], the KS Hamiltonian at time t, HKS [ρ(t), t] only depends on the
density at the same time ρ(t) and the time t itself. We only concern ourselves with
adiabatic TDDFT functionals, so this will remain true for the rest of this chapter.
The formal solution to the TDKS equations is:
φ(t + dt) = U(t + dt, t) · φ(t)
Z t+dt
U(t + dt, t) ≡ T exp{−i
HKS [ρ(τ )τ ]dτ }
(2.2)
t
Notice T , the time ordering operator, which reorders operators such that they
always go from later to earlier times from left to right.
HKS [ρ(t), t] is composed of three main parts,
HKS [ρ(t), t] = −
1 2
∇ + vext (t) + vJxc [ρ(t)]
2
the kinetic energy (− 21 ∇2 ) the external potential (vext ) and the inter-electron
Coulomb-exchange-correlation potential (vJxc ). Note that only vext has an explicit
time dependence. The kinetic energy operator is time-independent, whereas the
53
Coulomb-exchange-correlation potential is time-dependent only through the density
ρ(t). The implicit time-dependence of the Coulomb-exchange-correlation potential is
an important detail which we will revisit later.
The one-particle density matrix (1PDM) P(t) and the KS orbitals φi (t) are equivalent ways of representing the states of the single-determinant KS wavefunction, with
the 1PDM being slightly more general due to its ability to represent ensembles of
single-determinant states as well. The 1PDM evolves as such:
P(t + dt) = U(t + dt, t) · P(t) · U† (t + dt, t).
(2.3)
with U defined in 2.2. Note that P is the 1PDM of the Kohn-Sham reference
wavefunction, and not the 1PDM of the true wavefunction. We choose to use the
1PDM to represent the system both because it gives rise to more elegant equations
and also because for insulators the 1PDM in a spatially local basis is sparse, i.e.
the number of nonzero elements eventually scales linearly with system size for large
systems [128, 129, 130]. This would allow a linear-scaling implementation of real-time
TDDFT to be done in the future.
During a simulation, the total propagation interval [t, t + δT ] is broken up into
T /dt timesteps of size dt. For each time step there are three primary operations: 1)
constructing HKS 2) constructing U(t+dt, t) and 3) evolving the 1PDM using Eq. 2.3.
Sophisticated schemes may involve each of these operations multiple times and
in a different ordering. For the moderate-sized systems we apply our methods to,
the computational effort is dominated by the first step, that of computing the KS
Hamiltonian HKS. Under our scheme, HKS is computed a fixed number of times for
each timestep, and so efficiency is attained by reducing the number of steps T /dt
taken by increasing the size of each timestep dt. Step 2 and 3, under our implementation, are computed using standard computational dense linear algebra routines. To
obtain a linear-scaling protocol, one would need to replace all the linear algebra with
corresponding sparse matrix algorithms [131].
54
Time-Ordering
By dividing the propagation interval into timesteps, we assure the correct timeordering between different timesteps. However, as we use longer individual timesteps,
the time-ordering of operators within each timestep becomes important. This can be
accomplished by using the Magnus expansion for the time-ordered exponential T exp:
T exp{−i
Z
t
Z
t+dt
Â(τ )dτ } = exp(Ω̂1 + Ω̂2 + Ω̂3 + · · · )
t+dt
Ω1 = −i
dτ Â(τ )
t
Z t+dt
Z τ1
Ω2 =
dτ1
dτ2 [Â(τ1 ), Â(τ2 )]
t
t
Z t+dt
Z τ1
Z τ2
Ω3 = i
dτ1
dτ2
dτ3 ([Â(τ1 ), [Â(τ2 ), Â(τ3 )]] + [[Â(τ1 ), Â(τ2 )], Â(τ3 )])
t
t
t
..
.
where the subscript n in Ωn indicates the order to which the expansion is correct. For
example,
T exp{−i
Z
t+dt
t
Â(τ )dτ } ≈ exp(Ω̂1 + Ω̂2 + O(dt3 )) = exp(Ω̂1 + Ω̂2 ) + O(dt3 )
Discretization
We have, at this point, converted the time-ordered exponential into a a normal exponential of some integral terms. Next, we discretize the integrals so that they can
be implemented as a finite sum on the computer. Blanes et al. [132] showed that the
operators Ωn can be efficiently approximated to order O(dt2N ) using Gauss-Legendre
quadrature and evaluating the integrand Â(t) at N time points. The first order term
Ω1 is approximated as
55
Z
t+dt
Â(τ )dτ ≈
t
N
X
wi Â(τi ) + O(dt2N +1)
i
where wi and τi are the weights and grid points from the N th -order Gauss-Legendre
quadrature. Blanes et al. showed that, using the same grid points, τi , it was possible
to construct the higher order Ωn terms as well. For example, to second order,
τ1 = t + dt/2,
Âi = Â(τi ),
Ω̂1 ≈ −iÂ1 · dt + O(dt3 )
T exp{−i
Z
t+dt
t
Ω̂2 ≈ 0 + O(dt3)
Â(τ )dτ } ≈ exp(Ω̂1 + Ω̂2 ) + O(dt3 )
= exp(Ω̂1 ) + O(dt3 )
(2.4)
and to fourth order,
τ1 = t +
τ2 = t +
T exp{−i
Z
t+dt
t
√ !
1
3
· dt,
−
2
6
√ !
1
3
· dt,
+
2
6
Âi = Â(τi ),
dt
+ O(dt5),
Ω̂1 ≈ −i Â1 + Â2 ·
2
h
i √3dt2
Ω̂2 ≈ Â1 , Â2 ·
+ O(dt5 )
12
5
Ω̂3 , Ω̂4 ≈ 0 + O(dt )
Â(τ )dτ } ≈ exp(Ω̂1 + Ω̂2 + Ω̂3 + Ω̂4 ) + O(dt5 )
= exp(Ω̂1 + Ω̂2 ) + O(dt5 )
By evaluating the operator Â, in our case HKS, N times within the timestep, we
56
obtain a unitary approximation which is correct to O(dt2N ).
Predictor-Corrector
The implicit time-dependence of the U integrand is a significant problem for Magnus
integrators. There is a circular dependency which needs to be resolved. The objective
of each timestep is to find ρ(t + dt) from ρ(τ ) at τ ≤ t. This requires the evaluation
of U(t, t + dt), which in turn involves evaluating HKS[ρ(τ ), τ ] for τ ∈ [t, t + dt]. Since
we do not know ρ(τ ) for τ > t, this is a problem.
Previous applications of the Magnus propagator to TDDFT [121] used iterative
techniques to determine ρ(τ ) for τ > t. Unfortunately, such an approach requires
many, e.g. 5-10, HKS builds per timestep, and consequently increases the computational cost of each timestep significantly. Instead of iterating to self-consistency, we
use a predictor-corrector scheme:
1. Use extrapolation to predict HKS [ρ(τ ), τ ] for τ ∈ [t, t + dt] from HKS [ρ(τ ), τ ] for
τ <t
2. Use the approximate HKS [ρ(τ ), τ ] and ρ(t) to find ρ(τ ) for τ ∈ [t, t + dt].
3. The ρ(τ ) for τ ∈ [t, t + dt] are then is used to construct a corrected HKS[ρ(t), t]
for τ ∈ [t, t + dt].
4. The corrected HKS [ρ(t), t] for τ ∈ [t, t + dt] are used to compute U(t, t + dt),
which is then used to propagate the ρ(t) to ρ(t + dt).
The predictor step does not involve computing new HKS , and is hence inexpensive.
Also, since we are extrapolating integrands, the extrapolation only needs to be correct
to O(dt2N −1) for the Magnus expansion to be accurate to O(dt2N ). The predictor step
alone is either dissipative or unstable, and without the corrector step would eventually
fail. We use the Magnus expansion for the both the predictor and corrector steps,
taking full advantage of the radius of convergence of the Magnus expansion [133] at
a modest cost.
In detail, the second order algorithm is illustrated in Fig. 2-1. The component
steps are:
57
Figure 2-1: Predictor-Corrector routine for the 2nd order Magnus integrator. The
order row shows the time order (in dt) to which the matrices in the same column are
correct to.
1. (Predictor) HKS matrices stored from previous time steps, 1a and 1b, are used
to extrapolate HKS matrix 3 to order O(dt): HKS (3) = − 34 HKS (1a)+ 47 HKS (1b).
2. (Predictor) Using 3, the density matrix 2 is propagated to 4 using Eq. 2.4. This
is correct to O(dt2 ).
3. (Corrector) Density matrix 4 is used to compute the HKS matrix 5 .
4. (Propagation) HKS matrix 5 is used to propagate the density matrix 2 to density
matrix 6 using Eq. 2.4. This is correct to O(dt2 ).
5. (Update) For the next step, HKS matrix 1b becomes 1a, HKS matrix 5 becomes
1b, and density matrix 6 becomes 2. Other matrices are discarded, and the
process starts again from step 1.
We have also derived the analogous 4th and 6th order Magnus expansions using
Mathematica [134]. Since our results (see below) indicate these propagators are not
preferable unless relatively high accuracy is desired, we only present the (quite involved) 4th , 6th and 8th order predictor-corrector schemes in the appendix.
58
2.2.2
Numerical Validation
We implemented the Magnus integrators described above in a local version of the
program NWChem [135]. For simplicity, we assume the nuclei are fixed and that the
time dependence is generated by a (user-specified) sum of pulses of the form
2
(t−tk )
1
2
e 2σk cos(ωk t + φk )
v̂k (r, t) = αk Ôk (r) √
2πσk
where Ôk is an arbitrary operator. Our implementation uses the existing optimized
subroutines to construct HKS as needed. Since intuition and numerical experiments
dictate that this will be the rate limiting step, this allows us to easily produce efficient,
flexible code. Note that, for hybrid functionals like B3LYP [136], NWChem follows
the standard prescription of using the Hartree-Fock-like nonlocal form for vxc . This
is technically outside the domain of Kohn-Sham DFT - which requires a local vxc
- but we do not expect any significant errors from this well-tested approximation.
The predictor and propagation phases of the calculation are formulated in terms
of standard matrix operations (multiplication, inversion and diagonalization) in the
atomic orbital(AO) basis, which are inexpensive for systems with less than a few
thousand basis functions. For example, in the AO basis the propagator becomes
U(t) = exp(S−1 · Ω(t)) = S−1/2 · exp(S−1/2 · Ω(t) · S−1/2 ) · S1/2
where S is the AO overlap matrix. U(t) is computed exactly by: 1) Computing
S1/2 and S−1/2 (only needs to be done once for all times) 2) diagonalizing S−1/2 ·
Ω(t) · S−1/2 3) exponentiating S−1/2 · Ω(t) · S−1/2 in the eigenbasis and 4) pre- and
post- multiplying the result by S−1/2 and S1/2 . There are no approximations due
to Chebyshev expansions or Trotter factorizations of the propagator and thus all the
Magnus propagators are rigorously unitary.
Our first task is to determine what timesteps are appropriate for these Magnus
propagators. The value of having a large critical timestep is perhaps best illustrated
by plotting the wall time required to propagate the 1PDM with a given accuracy. In
59
Wall Time/Simulation Time (x10-15)
103
102
101 -12 -10 -8
10 10
10 10-6 10-4 10-2 100
Average Absolute Error
Figure 2-2: Minimum wall time required to obtain a prescribed average absolute
error in the final density matrix of methane (B3LYP/6-31G*, 120 a.u of propagation)
using various approximate propagators: 2nd order Magnus (green dashed), 4th order
Runge-Kutta (teal dot-dashed with squares), 4th order Magnus (red solid) and 6th
order Magnus (blue dotted).
Figure 2-2, we present an illustration of this type for methane in a 6-31G* basis [137]
using the B3LYP functional [136]. We begin with the molecule in its ground state
and apply a dipole pulse along one of the C2 axes with a Gaussian envelope in time
(intensity αk = .1, width σk = 5, center tk = 50, all in a.u.) and evolve the system
for 120 a.u. At the end of each simulation, we measure the error
Error =
1 X exact
|Pij (tf ) − Pijapprox (tf )|
2
K i,j
where K is the number of basis functions and the “exact” density matrix is obtained
using 6th order Magnus with a very small time step. It is clear from the figure that
for very high accuracy the higher order methods outperform the low-order methods.
If one was interested in obtaining near-machine precision results, the high order propagators are the clear choice. However, more typically, we are interested in the most
economical way to obtain results of reasonable accuracy (e.g. with errors of order 10−4
in the 1PDM). As can be seen in Figure 2-2, the low order methods are often more
efficient for moderate accuracy. While 4th - and 6th - order Magnus are always more
60
precise than 2nd order for a fixed timestep, 2nd order Magnus is often more efficient
(i.e. requires the least wall time) due to the fact that it requires one half (one third)
as many HKS builds as 4th (6th ) order Magnus. As a result, for moderate accuracy, 2nd
order Magnus can actually be the propagator of choice. We have also compared these
Magnus expansions to the commonly used 4th order Runge-Kutta (RK4) integrator.
RK4 abruptly diverges even for fairly small time steps of .2 a.u., resulting from the
loss of normalization in the Kohn-Sham wavefunctions (see Fig. 2-2). On the other
hand, Magnus integrators are convergent for every time step, and with these larger
timesteps, Magnus propagators are 15-20 times more efficient than RK4 for these
systems. The results from this and other test cases indicate that 2nd order Magnus
with a time step of 1-2 a.u. is usually the most efficient way to propagate the density
within an error of 10−4-10−5 .
2.3
Application: Conductance of a molecular wire
Figure 2-3: Schematic of the source-wire-drain geometry used in the present simulations. The bias is applied to the left and right groups of atoms, which act as a source
and drain for electrons, respectively. For different wire lengths (e.g. 50 carbons versus
100) the wire length is kept fixed and the size of the source and sink are varied.
We now outline how TDDFT can be used to compute the conductance of a short,
four-carbon segment of a polyacetylene wire where the role of the reservoirs is played
by the semi-infinite left- and right- strands of the wire . Now, as stated previously,
TDDFT is only able to handle the dynamics of closed quantum systems, so we must
make a finite model of the infinite wire if we hope to make any progress. We will
therefore focus our attention on oligomers of the form CN HN +1 −C4 H4 -CN HN +1 (see
Figure 2-3) with the implicit assumption that N must be chosen “large enough.”
Our choice of these model systems is inspired by previous work on short carbon wires
61
embedded in jellium layers [87, 110]. The resulting calculations on these simple wires
are intended to illustrate the important points that will arise for more complicated
junctions.
We will further restrict our attention to only one particular functional (B3LYP
[136]) and one particular basis set (6-31G* [137]). There are undoubtedly interesting
variations on these results with different model chemistries. However, our emphasis
here is on the changes that must be applied independent of the model chemistry in order to extract transport properties from the dynamics. Thus B3LYP/6-31G* simply
serves as a good model chemistry and could be replaced with any other combination
of functional and basis set. Further, since we are interested in treating elastic conduction, the nuclei are held fixed throughout each calculation at their optimal positions
at zero bias.
2.3.1
Voltage Definitions
We focus on potential-driven (rather than current-driven) conduction. In this case,
there are at least two different prescriptions one can use to simulate conduction using
TDDFT, depending on whether the current is driven by a chemical potential bias (µ)
or a voltage bias (V). In the V case, the TDDFT prescription is to begin with the
system at equilibrium with no external potential and then turn on a voltage VL (VR )
in the left(right) lead such that V=VL -VR . The resulting voltage bias will drive a
current from left to right for positive V. The µ case is somewhat more complicated.
Here, one considers that the system is connected to two reservoirs (L and R) that are
held at constant chemical potential (µL and µR ). If the leads are large enough, this
can be accounted for by equilibrating the system with each lead held fixed at its own
chemical potential (µL or µR ). Then, at time zero, the constraining chemical potential
is removed allowing current to flow from regions of high chemical potential to low.
Depending on the experiment, either scheme could be the more appropriate model,
but the µ- and V- biased prescriptions tend to give very similar I-V curves [138]. We
expect differences to primarily manifest themselves in the distinct transient dynamics
of µ- and V- biased junctions [139] and the convergence of the two prescriptions
62
toward the thermodynamic limit. Since both of these issues are germane to the task
of making the simulations “large enough” and “long enough” to mimic experiments,
we will examine both prescriptions in what follows.
No matter which bias scheme one chooses, there is necessarily some ambiguity
about how one defines the potential. The only piece of experimental information
we have is that, when averaged over a macroscopic volume deep in the leads, there
is a constant shift of the potential on the left relative to the right. This leads to
any number of different microscopic potentials that satisfy this condition. Step-like
potentials [97, 112], ramp potentials [94, 110], and potentials defined in terms of
localized orbitals [96, 98, 100, 101] all give qualitatively similar I-V curves. In this
paper, we propose to use atomic Löwdin populations [140] to define the potential in
the following way. First, we note that the Löwdin population (N X ) for a given set
of atoms (X) can be written as the trace of an operator matrix with the one particle
density matrix (P):
N X ≡ T rPWX
(2.5)
where WX is given by
WijX ≡
X
1/2
1/2
Si,α Sα,j
(2.6)
α∈X
and the summation runs only over atomic orbitals centered on atoms in the fragment
X. It has been shown that WX is a projection operator [141], so the atomic populations are always non-negative. Next, we define the the bias potentials VB (B = L, R)
to be simply a constant VB times the appropriate Löwdin operator:
V B ≡ VB W B .
This choice is motivated by previous studies within our group, which show that using
this population definition within constrained DFT leads to a consistent treatment of
long-range charge transfer excited states [142, 19] and low-lying spin states [143].
63
Given this definition of the potential, both the µ- and V -biased cases can be
simulated using an appropriate time dependent potential V̂ (t). For the chemical
potential, V̂ (t) = VB θ(−t), while for the voltage bias, V̂ (t) = VB θ(t) where θ(t) is
the Heaviside step function. In both cases, the system is equilibrated to the ground
state at t = −∞ (or, equivalently, at time t=−ǫ) and then propagated forward in
time using TDDFT. Because the system always begins in the ground state, there
are no ambiguities about initial wavefunction dependence in either case. We note
that the function used to turn the potential on or off in time is arbitrary. However,
the average currents presented below are insensitive to the choice of the switching
function as long as f (t) changes from 0 to 1 within ≈ 15 a.u. Slower switching results
in a partial depletion of the finite reservoirs before the bias is completely established.
In principle, the current through the device also needs to be defined. In the
experiment, the current is measured deep in the leads and in a finite system it is
not clear where the dividing surface should be placed in a simulation to best mimic
the experiment. However, because we have chosen to define our bias in terms of
a particular (albeit arbitrary) population definition, the definition of the current is
uniquely determined via the continuity equation:
1
2
1
I~ · ~ndσL −
2
L
Z
d (N L − N R )
I~ · ~ndσR =
dt
2
R
Z
(2.7)
where σR and σL are the surface elements associated with the boundaries of the left
and right leads. The left hand side is the current we seek: the average of the current
out of the left hand lead (first term) and the current into the right hand lead (second
term). The surface implied by the use of Löwdin populations is extremely complicated
to define, and hence the left hand side is extremely difficult to evaluate. On the other
hand the right hand side is just the time derivative of the Löwdin populations, easily
obtainable from TDDFT. Further, since our bias couples directly to N L and N R
it is most natural to think of the fluctuations in these variables as generating all
the dynamics. For the present case, we use the right side of Eq. 2.7 to define the
current through the junction in our simulations. Indeed, for any given definition of
64
the potential, Eq. 2.7 gives a unique prescription for the current through the device
region. This equivalence is part of a deep connection between current and number
fluctuations in electrical junctions [144].
2.3.2
Current Averaging
Figure 2-4: Initial density corresponding to a chemical potential bias in polyacetylene.
Red indicates charge accumulation and green charge depletion relative to the unbiased
ground state. At time t=0 the bias is removed and current flows from left to right.
µ = 6.80 V
µ = 5.44 V
µ = 4.08 V
µ = 2.72 V
µ = 1.36 V
Current (mAmp)
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
500
1000 1500 2000 2500
Time (Attosec)
Figure 2-5: Transient current through the central four carbons in C50 H52 at a series
of different chemical potential biases. There is an increase in current as voltage is
increased, along with large, persistent fluctuations in the current. The currents are
converged with respect to time step and the apparent noise is a result of physical
fluctuations in particle flow through the wire.
We first study the molecular wire C50 H52 under a chemical potential bias. The
electronic energy of the molecule is minimized while the left (right) C23 H24 segments
are subject to a bias of +µ (-µ) in the Löwdin potential. An example of one such initial
state is illustrated in Figure 2-4 for a short wire. At time zero, the bias is removed and
65
µ = 6.80 V
µ = 5.44 V
µ = 4.08 V
µ = 2.72 V
µ = 1.36 V
Current (mAmp)
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
500
1000 1500 2000 2500
Time (Attosec)
Figure 2-6: Transient current through the central four carbons in C50 H52 at a series of
different chemical potential biases smoothed over a time window of width ∆t = .36f s.
The average currents are now move clearly visible. The slow decay of the current at
later times results from the partial equilibration of the finite left and right leads.
the electrons are allowed to relax. Figure 2-5 shows the resulting current at a series of
different voltage biases. Several important points follow from this figure. First, even
for this wire length, the left and right chemical potentials equilibrate very rapidly,
as evidenced by the fact that the current pinches off within about 2.5 fs. Second,
there is clearly a general trend toward increasing current as we increase the bias,
reflecting the current-voltage relationship for this wire. Finally, significant transient
current fluctuations (‘noise’) hinder the identification of the average current based on
in these calculations.
The noisiness of the data can be overcome by realizing that experimental measurements are made on a much coarser timescale than the timestep of our simulation. A
better approximation to the experimental current can be made by averaging Eq. 2.7
over a relatively wide time interval ∆t:
Iavg ≡
(N L (t) − N R (t)) − (N L (t − ∆t) − N R (t − ∆t))
.
2∆t
(2.8)
This expression physically corresponds to the gedanken experiment where an apparatus with finite time resolution ∆t checks the number bias twice in succession and
66
then uses the mean value theorem to approximate the derivative. This process will
ignore fluctuations that occur on a timescale faster than ∆t resulting in qualitatively
smoothed current profiles. Figure 2-6 shows the transient currents obtained when
one applies Eq. 2.8 (with ∆t = .36f s) to C50 H52 at various biases. As expected, the
transient fluctuations are suppressed and one can now see the earmarks of smoothly
increasing current in these molecular wires. It is somewhat remarkable that these
molecules are able to attain a quasi-steady state so quickly (faster than 1 fs), and
similar observations have been made previously for a simple gold wire [112]. We attribute this fast relaxation in molecular wires to the nearly perfect coupling between
the “leads” and the “wire”. Strong system-bath coupling leads to a very short lifetime
Max. Smoothed Current (mAmp)
for transient states and quick relaxation.
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
1
2
3
4
5
Chemical Potential (V)
6
Figure 2-7: Maximum smoothed current through the central four carbons in C50 H52
as a function of chemical potential bias (red pluses). For comparison, we also present
the analogous result for the central carbons in C100 H102 (green squares) demonstrating
convergence of the calculation with respect to lead size. The blue line is a linear fit to
the C50 H52 data at low bias indicating that polyacetylene is an Ohmic resistor with
a conductance of ≈ .8G0
If we interpret the maximum smoothed currents in Figure 2-6 as the appropriate
steady-state current for each voltage, we obtain the current-voltage relation shown
in Figure 2-7. From this graph it is clear that the current through the wire increases
in very nearly linear fashion over a very wide range of voltages. There is a leveling
off in current at large voltages that results from an essentially complete depletion
67
of the valence states of the central part of the wire. Perhaps surprisingly, while the
conduction is quasi-Ohmic, the conductance is not one quantum of conductance (G0 ≈
74.5µS). Instead, the low-bias behavior is better approximated by a conductance of
.8 G0 . Initially, one might suspect that this is a finite size effect; that is to say that if
the leads were “long enough” the conductance would approach G0 . To check this we
have run similar calculations changing the size of the leads in the wire, keeping our
attention on the conductance of the central four carbons. As an example, Figure 2-7
also shows a few I-V points for C100 H102 . We have considered only a few voltages
because of the increased computational burden per voltage point. However, even this
sparse set of data allows us to conclude that the differences between the C50 and
C100 data are quite small and have little influence on the conductance. Thus, our
approach of using larger and larger finite systems has indeed converged to the open
system limit, but with a finite wire resistance.
In order to understand the origin of the conductance in this wire, we return to the
original Landauer picture. Because the “leads” have an essentially perfect connection
with the “molecule” in these polyacetylene wires, the appropriate picture involves
strong coupling between the leads and the molecule. This coupling broadens the
molecular levels to the point where the I-V curve becomes featureless. Recall that,
in the weak coupling limit, the current displays a staircase structure as a function of
bias, and so Figure 2-7 should be interpreted as the limiting case where this staircase
pattern is “smeared out” yielding a smooth I-V characteristic. Thus, the conductance
of the central four carbons should not be viewed as coming from a single quantum
state, but from the superposition of a number of broadened states. The numerical
value of the conductance thus reflects the density of states available in this wire.
2.3.3
Comparison to NEGF results
Assuming that the dynamics described so far approximate the steady state of the
infinite system limit, the present approach is equivalent to the NEGF method [106].
Thus, as a test, we can compare the real-time DFT results with those obtained via
DFT-NEGF. We use Lorentzian broadening on the finite lead to simulate the infinite
68
0.8
Current (mAmp)
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
1
2
3
4
Bias (V)
5
6
Figure 2-8: Maximum smoothed current through the central four carbons in C50 H52
as a function of chemical potential bias using real-time TDDFT (top red line) and an
NEGF approach described in the text (bottom blue line). The two calculations are
nearly identical at low bias and differ somewhat at higher biases due to the lack of
self-consistency in the NEGF results.
size limit [99]. We do this by first using Löwdin’s symmetric orthogonalization [140]
on HKS, partitioning the molecule in exactly the same way as the real-time TDDFT
simulation. We then add a small imaginary component, ǫ = .055, to the eigenvalues
of the orthogonalized HKS matrix, broadening the energy distributions of the lead
states so that they now resemble the continuum present in infinite-sized leads. The
value of ǫ was set by maximizing the current, consistent with our practice of picking
the maximum current from the smoothed real-time TDDFT trajectories. Using the
broadened density of state profile, we then computed the current using techniques
outlined in Ref. [99]. The NGEF approach we use is not self-consistent, because HKS
is computed using the ground-state electron density. However, the results obtained
should be comparable to real-time TDDFT insofar as the self-consistent density resembles the ground-state density at low biases.
The NEGF results obtained for C50 H52 are presented in Figure 2-8, alongside realtime TDDFT results from the previous section. The two techniques agree almost
completely at low bias, and give qualitatively similar results at larger biases. Much of
the difference can be attributed to the lack of self-consistency in the NGEF procedure
69
- at greater biases, the self-consistent density deviates more from the ground-state,
leading to greater differences between self-consistent and ground-state results.
These results support our assertion that the real-time TDDFT simulations are
informative of the open-system limit - enlarging the leads does not change the conductance curve (Fig. 2-7), and the results are consistent with NEGF calculations
(Fig. 2-8) in the low-bias limit. We are hence able to conclude that even at relatively
small lead sizes, our system already reflects the electron transport dynamics present
in the large lead limit. This gives us reason to believe that a simulation of a metalmolecule-metal with finite metal leads would also be relevant to the experimental
infinite metal lead setup. As we explain later in this chapter, however, we anticipate
additional challenges for this latter case because of the physical relevance of noise in
the simulation.
Voltage Biased Case
Max. Smoothed Current (mAmp)
2.3.4
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
1
2
3
4
Bias (V)
5
6
Figure 2-9: Maximum smoothed current through the central four carbons in C50 H52
as a function of chemical potential bias (top red line) and voltage bias (bottom green
line). The results are quite similar until 4 V, at which point the bias is so large
that the finite width of the valence band for polyacetylene causes the conductance to
plateau in the voltage biased case.
The preceding results were all obtained using a chemical potential bias. We have
also investigated the effects of using a voltage bias to generate the conductance and
find that the results are qualitatively no different. To illustrate this point, the current70
voltage plot for C50 H52 using both µ and V biases is shown in Figure 2-9. Clearly,
the differences between the two schemes are small until a bias of about 4 V, at which
point the voltage biased results show some negative differential resistance. A brief
inspection reveals that the plateau at high bias again results from the finite width of
the polyacetylene valence band, which apparently has a somewhat larger influence in
the voltage biased case.
2.3.5
Transient Fluctuations
µ = 6.80 V
µ = 5.44 V
µ = 4.08 V
µ = 2.72 V
µ = 1.36 V
Current (mAmp)
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
500
1000
1500
Time (Attosec)
2000
Figure 2-10: Transient current through the central four carbons in C100 H102 at a series
of different chemical potential biases. The current fluctuations previously observed
with smaller reservoirs in C50 H52 persist and are therefore not associated with a finite
size effect.
Note that the fluctuations in the current shown in Fig. 2-5 are not numerical noise.
These fluctuations are present even at smaller timestep sizes, and reflect deterministic
variations of the current in the wire. Furthermore, the fluctuations are also converged
with respect to lead size, and not due to the finite size of the leads. For example,
examining the current fluctuations with larger reservoirs in C100 H102 one notices that
the magnitude of the variations in the current apparently approach a constant value
rather than falling to zero (See Figure 2-10). This fact indicates that these fluctuations
are characteristic of some physical current noise in the wire. We can easily quantify
this noise from the transient current traces shown in Fig. 2-10 by simply computing
71
0.08
Noise (x 10-3a.u.)
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
Current (x 10-2a.u.)
Figure 2-11: Statistical noise in the current through the central four carbons in
C100 H102 . The data (squares) can be fit to a sub-Poissonian distribution (line).
the statistical uncertainty:
S(I) =
Z
ˆ − Ii
¯ 2 dt
hI(t)
(2.9)
where I¯ is the average current for the given voltage and the integration runs over
the quasi-steady state currents shown in Figure 2-10. We find that it is somewhat
more difficult to converge the noise than the conductance - somewhat longer time
windows, and hence somewhat longer wires, are required to obtain accurate noise
compared to the conductance. This is consistent with the experimental situation,
where successively higher moments of the electron counting distribution are progressively more difficult to obtain [145]. However, if we average over a window of 70 a.u.
(≈ 1.7 fs), we obtain reasonably stable results for the noise in C100 H102 , as shown
in Figure 2-11. Clearly the noise increases linearly with increasing current, but the
slope is much less than 1, indicating a sub-Poissonian process. It is not clear what
the origin of this noise is. It may be that the application of the bias leads to a finite
population of one or more molecular excited states which then keeps the system from
relaxing to a steady state. Alternatively, this could be the result of the interference of
incoming and outgoing waves within the molecule. We have not successfully isolated
72
the origin of these fluctuations, and a case can be made for either picture.
Even without a molecular interpretation, the noise does give us some important
clues about real time transport simulations. There is a large body of work that
discusses the rich history of conductance noise in quantum transport [146, 147, 148,
149, 145, 150, 151, 152]. In particular shot noise describes the instantaneous, quantum
fluctuations of the current about its mean value:
S≡
Z
¯ 2 i.
h(I(t) − I)
(2.10)
where the average h...i is taken over a time long compared to the characteristic time
¯ Despite the mathematical similarity, the temporal noise (Eq. 2.9) is not
τ0 ≡ e/I.
shot noise (Eq. 2.10). Shot noise arises from quantum uncertainty of the current for
identically prepared systems, whereas our simulations reflect temporal uncertainty in
the average current. However, for the ideal case of single-electron transport through
a junction of transmittance T , the shot noise is is also sub-Poissonian [146, 147]:
S ≡ e(1 − T )I¯
If we use this expression to fit the temporal noise in polyacetylene, we obtain T ≈ .98,
as compared to the numerically observed conductance of .8G0 . This discrepancy is
to be expected on two grounds. First, because of the finite timestep, our simulations
actually reflect the noise power over a particular interval rather than the noise itself. Second, shot noise involves two particle correlations while the noise we consider
involves fluctuations in a one particle observable. These two quantities are in fact related via the fluctuation-dissipation theorem [153]: by considering the response of the
system to all possible potential perturbations, two particle observables such as hIˆ2 i
can be obtained without further approximation. By restricting ourselves to spatially
uniform bias potentials, we recover only a fraction of the associated fluctuations and
observe an attendant reduction in the noise. Hence the analogy with shot noise can
only be used for qualitative analysis of the temporal fluctuations in our simulations.
A characteristic of shot noise is its tendency to increase as transmittance is re73
duced. This suggests that for molecular junctions, which typically have much lower
conductances (typically .01G0 ), the temporal noise will be much more significant. In
contrast, a purely metallic wire would experience no shot noise in the large lead limit
because it is a perfect conductor. This explains the observation that for a gold wire
[112] the current-versus-time plots smoothly approach a plateau, while for polyacetylene, which only reduces the conductivity by 20%, noise already becomes important.
Thus, it seems likely that the transient currents in a .01G0 junction may be dominated
by shot noise over quite large time intervals, making the interpretation of real-time
simulations much more challenging.
2.4
Conclusions
In this chapter, we have demonstrated the feasibility of performing real-time TDDFT
simulations of the transient current dynamics through a molecular wire at finite bias.
These results were made possible by an efficient algorithm for integrating the TDKS
equations in a localized basis that allows us to take very large time steps. Taking polyacetylene as our prototypical example, we have shown that one can make a consistent
definition of current and bias within TDDFT and obtain quantitative predictions of
the current at finite bias by averaging the instantaneous current over a microscopic
time window. Further, we have verified that these currents are converged with respect to the size of the reservoirs and that the results agree with the corresponding
NEGF current-voltage character. Finally, we have investigated the temporal current
noise and shown that for these wires, the noise does not decay to zero as the wire
length is increased, but approaches a finite value. We argue that this noise is likely
to be more significant in junctions that have low transmittance, which implies that
real-time conductance simulations of MMM junctions -which have much lower transmittances than an isolated wire - may be quite challenging. In the near term, we plan
to use TDDFT to study the conductance properties of some simple MMM junctions.
The translation of this framework to such devices is straightforward in principle: one
constructs a supersystem that contains enough metal atoms to mimic the ’bulk’ properties of the leads and propagates the system under an appropriate bias until a reliable
74
average current can be extracted. Novel challenges that will need to be addressed
for MMM junctions include the sensitivity of SCF convergence for metallic systems,
the uncertain structure of the metal-molecule interface and the expected increase in
noise due to the low transmittance of the junction. The points raised here – efficient
integration of the TDKS equations, consistent definition of the bias potential, the
importance of smoothing to obtain the relevant current – will facilitate study of these
more technologically relevant devices.
For our time-dependent simulations, the average current provides an important
validation of the model; by comparing to other predictions of the DC current, we can
verify that our simulation is “large enough” and “long enough”. However, once the
model has been validated, the average current is only one of a host of properties a realtime simulation gives access to. Electroluminescence, finite bias impedances, driven
rectification and current triggered molecular dynamics can all be treated within the
framework described here. Hence, this “microcanonical” picture of electron transport
dynamics opens up a huge array of physical processes for theoretical study and we
are currently in the process of analyzing some of these effects. In particular, it
may be possible to predict the shot noise in a molecular junction without further
approximation using TDDFT in conjunction with the fluctuation dissipation theorem
[154].
Finally, in regards to current voltage predictions, it has been pointed out that there
are significant errors in the DFT currents at low bias due to improper cancellation
between self-interaction errors [109] and static electron correlation [93, 100]. We have
shown that for time-independent problems, the balance between static correlation
and self-interaction can be controlled by applying physically motivated constraints to
the electron density [142, 143, 19]. That approach, which we have dubbed Configuration Interaction Constrained Density Functional Theory, is described in the next
chapter. The analogous time-dependent approach, which constructs reference states
using constraints, and represents the time-evolution of the system within the space
of those states, could be used to study conductance in the future.
75
76
Chapter 3
Dissociation Curves using
Constrained Density Functional Theory
Configuration Interaction
Note: The bulk of this chapter has been published in Ref. [1].
3.1
Introduction
Electron correlation is the difference between an exact treatment of electronic structure and the physics present in Hartree-Fock, where mean-field inter-electron Coulomb
repulsion and Pauli exclusion have been accounted for. It affects important observables including ground state geometries, reaction barriers and dissociation energies.
Diatomic molecules experience a varying ratio of static to dynamic correlation
depending on how long the interatomic distance is. At distances close to the equilibrium, the electrons are at a close distance to each other and experience strong
Coulomb repulsion - this corresponds to dynamic correlation. At distances far from
equilibrium, near the dissociative limit, the near-degeneracies in energies lead to significant static correlation. Correlation energies in diatomic molecules are dominated
by one of the two forms of correlation depending on distance of separation, making
dissociation curves ideal for contrasting dynamic correlation with static correlation.
DFT is able to treat dynamic correlations well, but tends to fail when static correlation becomes important. In this chapter, we develop a method where constrained
DFT (CDFT) is used to construct an active space where dynamic correlation has
77
been accounted for, within which a new solution is constructed thus adding in static
correlation. We demonstrate the efficacy of the method by computing dissociation
curves for a few small diatomics, where we contrast our approach with the failures
of conventional DFT near dissociation [155]. We show that our method is able to
accurately treat both dynamic and static correlation dominated regimes of the dissociation curve more efficiently than wavefunction-based methods. The difference in
efficiency is significant as it makes future treatment of bond breaking and formation
in larger systems possible.
3.2
State of the Art
Under ab initio wavefunction-based methods, static correlation can be treated by using multiple reference (MR) methods [11], which use a multiple determinant reference
state, and dynamic correlation can be treated by the use of perturbation theory on
an uncorrelated single-determinant reference state [156, 157].
Density functional theory is in principle capable of treating both forms of correlation, but in reality DFT approximations deal with dynamic correlation a lot better
than static correlation [50]. It is for this reason that DFT tends to have good accuracy
[158, 159, 10] when dynamic correlation dominates, i.e. near equilibrium geometries,
and fares worse at geometries far from equilibrium.
Cases where most DFT functionals fail to describe dissociation correctly include
H+
2 , where it erroneously predicts a barrier to dissociation / association [160]. DFT
also predicts the dissociation of LiF into partially charged atoms, whereas in reality it
dissociates into neutral atoms [161, 162, 163]. In the case of neutral H2 , unrestricted
DFT gets the energy correct but with a broken-symmetry spin density, instead of the
correct symmetric spin density of a pure singlet [164, 165].
Improvements have been attempted while staying within the DFT formalism. It
is DFT’s ability to compute dynamic correlation cheaply that makes it attractive as
a starting point for methods attempting to treat both types of correlation. Staying strictly within DFT, there are many ways in which multireference character is
introduced, including the use of non-integer occupation numbers [166], an ensem78
ble of single-determinants [167, 168] and time-dependent density functional theory
[169]. There have also been attempts to combine DFT with wavefunction-based
MR methods like MCSCF [170], CASSCF [171], VB [172] and MRCI [173, 174]. A
key difficulty with mixing wavefunction and DFT-based methods is that of avoiding
double-counting, since there is no clear separation between dynamic and static correlation terms. There are also issues with using a density functional reference. Firstly,
the KS orbitals provided by DFT have no physical significance [40, 41, 42, 126], and
using them as a reference for perturbation is an uncontrolled approximation. Secondly, the accuracy of DFT is often attributed to the cancellation of error afforded
by many of the approximations - in this case, the effect of missing static correlation
is cancelled in part by the erroneous self interaction error (SIE) of DFT functionals.
If we were to treat static correlation more accurately without accounting for SIE, the
results would likely be worse.
3.3
Our Method: Constrained Density Functional Theory
Configuration Interaction (CDFT-CI)
We formulate a new method in the configuration interaction (CI) tradition, using
CDFT to construct an active space within which we compute the matrix elements of
the full Hamiltonian, which we then solve to arrive at our final answer. We call this
approach Constrained Density Functional Theory Configuration Interaction (CDFTCI). The active space is picked in a way as to, one, ensure that important configurations have been covered, and two, ensure that static correlation and SIE are both
minimized in the individual basis states. The states we have identified for this purpose
are localized electronic states as defined via CDFT, technology previously developed
within the group [142, 19], and described in the introduction chapter. These states
are prime candidates for an active space because they experience both little SIE and
little static correlation, and, as we will show later, can be systematically enumerated
to form a complete space by allusion to Valence Bond (VB) theory [175], all without
reference to orbitals.
79
3.3.1
Identifying an Active Space
As in other CI methods, we begin by identifying the active space which we use to
represent the electronic state. The choice of active space is informed by previously
formulated systematic approaches for finding energetically significant configurations,
and also by the knowledge of deficiencies in existing DFT functionals.
There are a number of CI methods in wavefunction-based electronic structure, and
they all come with prescriptions for picking an active space. Unfortunately, most of
those prescriptions call for delocalized molecular orbital configurations as a basis, and
CDFT does not do well for such configurations. There is, however, an exception in the
case of VB theory [175]. VB theory identifies configurations which resemble ground
states subject to spatial and spin density constraints, making them ideal for CDFT-CI.
Moreover, it has been shown that active spaces under CASSCF can be systematically
transformed into VB active spaces [176, 177]. For example, the CASSCF(2,2) active
space for single bond breaking consists of the configurations |σσ̄i, |σσ̄ ∗ i, |σ ∗ σ̄i and
|σ ∗ σ̄ ∗ i, which corresponds to the VB active space with ionic terms |LL̄i, |RR̄i and
covalent terms |LR̄i, |RL̄i. These VB configurations can easily be identified with
CDFT states that constrain spin charge density. Exploiting these facts, we adopt
the following prescription for identifying CDFT-CI active spaces. First, we find the
appropriate CASSCF(n,m) active space for the system we are studying. Second,
we transform that space with a unitary transformation into a number of VB theory
configurations. We then find the spatial and spin density constraint that corresponds
to each VB active space state, and thus define the CDFT-CI active space. This
prescription allows us to adapt the systematic enumeration of configurations that
CASSCF offers to CDFT-CI. Note that when m and n are big, there may exist
CASSCF(n,m) configurations that do not correspond to any CDFT constraint. In
such cases, we have not solved the problem of finding a CDFT-CI active space. In
the applications that follow, however, the procedure as defined here is sufficient.
As might have been noted, we go to considerable effort to obtain localized configurations for our active space. We are motivated to do so for two main reasons.
80
One, due to self-interaction error, available density functionals tend to underestimate
the relative energy of fractionally charged states compared to integer charged states
[47, 48]. This effect is best seen when an electron is delocalized between distant nuclei, like in the case of stretched H+
s . Not only is there a large error in the energy of
such radical dissociations [160], but that error also gets worse with larger distances
[178, 179]. In the case of neutral, closed-shell molecules, the erroneous SIE can cause
dissociated states to have the wrong density [161, 162, 163]. We are interested in
describing dissociation accurately, and so it is important to avoid SIE by using localized configurations. Two, when stretching a singlet diatomic under Kohn-Sham
DFT (KS-DFT), the dissociation energy obtained via common functionals misses the
strong static correlation component present in the stretched system. In contrast to
the KS-DFT ground state, broken-symmetry localized configurations have less static
correlation and can be accurately treated by DFT [165]. Taking advantage of this
fact, we use constraints to break the spin symmetry in each of our localized configurations, and then restore the right spin symmetry through configuration interaction
later on.
The explicit enumeration of configurations has a drawback for treating extended
systems. Even when the added molecules do not interact with the existing ones,
they add multiplicatively to the active space basis. Let’s say a molecule requires N
active configurations to describe - to represent two such molecules placed far apart
from each other, one would need to account for every possible combination of those
N configurations, leading to the need to consider N 2 configurations. If one wishes
to enforce size-consistency, the exponential growth of complexity with system size is
not resolved with the CDFT-CI approach. However, the severity of the problem is
reduced by the efficiency of DFT.
3.3.2
Computing Matrix Elements
Defining constraints for each configuration
wc in Eq. 1.22 assigns part of the electron density measured at each spatial point to
molecular fragment C, which is defined as a set of atomic centers. The sum total of
81
the densities assigned to every fragment results in the total density. For consistency,
we require that our definition of wc be the same throughout the CDFT-CI procedure.
Previous work by our group [19, 20] showed that Becke’s spatial weight functions
[18], which assigns density to atomic centers, are a good choice for the atomic density
assignment, and we use that definition here.
When fragments are far apart from each other, it is easy to assign Nc - one
simply counts up the number of electrons in the constrained region, which due to the
separation is usually a well-defined integer. When the individual fragments are close
to each other, however, it is no longer so clear how many electrons should be in each
fragment. In such cases, we use the following procedure to define Nc .
Depending on the configuration we are interested in, we first compute a reference
promolecule density ρ0 from a combination of molecular fragment densities
ρ0 (r) =
X
ρa (r).
(3.1)
a
where index a runs over the individual molecular fragments. The choice of fragment charges identifies the configuration we are trying to obtain. For example, for a
molecule consisting of two fragments A and B, the promolecule density corresponding to the ionic configuration A+ B− would be the combination of the A+ and the B−
densities, while the promolecule density corresponding to the covalent configuration
A↑B↓ would be the result of combining densities from the two open-shell fragments
A↑ and B↓ that have Ms = ± 12 respectively. The atomic fragment densities we use
are always spherically averaged.
Given a reference promolecule density ρ0 , we then constrain the expectation of
all the fragment density operators wc to give the same constraint value as in the
promolecule density:
Z
i
wC (r)ρ (r)dr −
Nic
=
=
Z
Z
i
wC (r)ρ (r)dr −
Z
wC (r)ρi0 (r)dr
wC (r)(ρi (r) − ρi0 (r)) = 0,
82
(3.2)
+
=
B
A
AB
=
+
A−
B+
A− B +
Figure 3-1: Figure taken from Ref. [1] Obtaining effective constraint value from
promolecular densities. The constraints are on Fragment A. Integration over the
striped area will give NCeff .
where the index i runs over the different configurations. Thus we determine the
constraint values Nci . When the separation between fragments is large, the constraint
wC would only count density from molecular fragment C, and Nci would correspond
to the integer number of electrons in that fragment. When the different fragments
overlap, however, wC would also count density from neighboring fragments in the
promolecule density and a non-integer constraint value Nci would be obtained. Note
that constraints defined in this way are continuous functions of interatomic distances
- this is important for potential future work with force calculations. Later in this
chapter, in the section on LiF, we test the sensitivity of our results to our definition
of wc by using another population definition.
Computing Matrix Elements
Each of the configurations is the result of enforcing constraints as in Eq. 1.22 with
{wc , Nci }. Constraints are enforced using Lagrange multipliers Vci , so the resulting
P
state |ii is an eigenstate of the Hamiltonian H = H0 + c Vci wc with the eigenvalue
P
Ei + c Vci Nci where H0 is the unconstrained Hamiltonian. Matrix elements in the
basis of constrained states are computed as below:
83
Hij = hi|H0|ji
X
X
1
(hi|H0 +
Vci wc −
Vci wc |ji
=
2
c
c
X
X
j
+ hi|H0 +
V c wc −
Vcj wc |ji)
c
c
X
X
1
=
((Ei +
Vci Nci + Ej +
Vcj Ncj )hi|ji
2
c
c
−(Vci + Vcj )hi|wc |ji)
(3.3)
where i and j are symmetrized to ensure Hermiticity of the resulting Hamiltonian.
Two types of terms remain undetermined in Eq. 3.3 - the wavefunction overlap hi|ji
and the coupling term from the constraint operator hi|wc |ji. These terms cannot
be computed from DFT, and must be approximated. We use the Kohn-Sham singledeterminant wavefunction to compute these terms approximately - this approximation
has previously been used in the group to compute electron transfer couplings [20],
and is a natural choice for the case at hand.
For example, given two basis states
|Ψ1 i = |φ11 , φ12 , φ13 |
|Ψ2 i = |φ21 , φ22 , φ23 |,
1
hφ1 |S|φ21 i hφ11|S|φ22 i hφ11 |S|φ23 i
hΨ1 |S|Ψ2 i = det(σ) = hφ12 |S|φ21 i hφ12|S|φ22 i hφ12 |S|φ23 i
1
2
1
2
1
2
hφ3 |S|φ1 i hφ3|S|φ2 i hφ3 |S|φ3 i
hΨ1 |wc |Ψ2 i =
X
ij
(−1)i+j hφ1i |wc |φ2j iMij
where Mij is the determinant of σ with row i and column j removed.
84
Computing the Energy
This is the final CI step, and simply consists of diagonalizing the Hamiltonian Hij obtained previously. The basis is non-orthogonal, so the generalized eigenvalue equation
has to be solved.
|ΨCI i =
X
i
Ci |Ψi i
Hij Cj = ǫSij Cj
The ground-state energy ǫ is the smallest root of the secular equation
det(Hij − ǫSij ) = 0
3.4
Application
Unless otherwise stated, calculations were all done in the 6-311G++(3df,3pd ) basis
set with the BLYP [180, 181] and B3LYP [38] functionals in a developmental version
of Q-Chem [182]. Energies depicted in dissociation curves are zeroed at the value
obtained from summing fragment energies computed with the same basis set and
functional. The following notation is used to indicate the combinations of schemes
and functionals computed - for the exchange-correlation functional B3LYP, RB3LYP
and UB3LYP would be the restricted and unrestricted Kohn-Sham DFT results,
respectively, and CB3LYP-CI would be the CDFT-CI result. CDFT calculations are
always done in an unrestricted manner, with any resulting restricted ground-state
only appearing in the final CI step.
3.4.1
H+
2
This system has only one electron, and is thus treated exactly by Hartree-Fock. As
shown in figure 3-2, however, with both BLYP and B3LYP, Kohn-Sham DFT fails
to capture even qualitative aspects of the potential curve, predicting a barrier at
around 3 Å with the energy decreasing as a function of distance after the barrier.
85
This particular artifact is a well-known failure of DFT [160] and is blamed on selfinteraction error [47, 48]. B3LYP performs slightly better than BLYP because of its
exact exchange component, but still exhibits the same failure overall.
For this case, we used the minimal active space CAS(1,2), which prescribes the
two active configurations H+ H and HH+ . At long distances, the CDFT-CI solution
localizes the charge, making the resulting configuration simply that of a neutral atom
and a proton, which is treated correctly by DFT. At short distances, the coupling
term increases and gives the right mixture of the two configurations to reproduce the
correct ground-state energy. As can be seen in Fig 3-2, this accuracy persists into
the hard wall region. The promolecule definition for Nic is very important to this case
because there is only one electron - requiring any local constraint to integrate to one
would mean requiring the density to be zero everywhere else, and be unphysical to
enforce.
Since H+
2 does not have static correlation, this case demonstrates CDFT-CI’s
ability to fix SIE in DFT functionals. Even though exact exchange is not an explicit
part of CDFT-CI, the local states used as a basis ensure that SIE stays constant at
all distances, leading to an effective cancellation of error if we are to concern ourselves
only with energy differences.
60
Exact
CB3LYP−CI
CBLYP−CI
B3LYP
BLYP
Binding Energy (kcal/mol)
40
20
0
−20
−40
−60
−80
1
2
3
4
5
R (Å)
Figure 3-2: Figure taken from Ref. [1] Potential curves of H+
2.
86
3.4.2
H2
For this system, we use the aug-cc-pVQZ basis set and only present the B3LYP results because the BLYP results are largely identical. The potential curves for both
restricted and unrestricted B3LYP are presented in Fig. 3-3. Unrestricted B3LYP
gets the geometry and atomization energies correct, and the potential from unrestricted B3LYP dissociates correctly, albeit with a broken symmetry spin density
[164, 165]. Restricted B3LYP preserves the correct spin symmetry, but dissociates to
the wrong energy limit. The restricted and unrestriced B3LYP results diverge at the
Coulson-Fisher at about 1.6 Å.
For this case, we use the CAS(2,2) active space transformed to the four localized
states H↑H↓, H↓H↑, H+ H− and H− H+ . Using CAS(2,2), CDFT-CI dissociates H2
correctly, and displays the correct spin symmetry for the ground state, being an
equal mixture of H↑H↓ and H↓H↑.
CDFT-CI once again gives an accurate potential at all distances, doing no worse
than restricted B3LYP at equilibrium bond lengths, and outperforming it at greater
bond lengths. For a clearer comparison of unrestricted B3LYP and CDFT-CI B3LYP,
we compare the errors from both directly in Fig. 3-4. The exact answer is obtained from a full CI calculation. The unrestricted B3LYP error has a maximum
of 5 kcal/mol at 1.8 Å, while its CDFT-CI counterpart has a more slowly varying
error which peaks no larger than 3 kcal/mol. For this case then, CDFT-CI is a better
method for studying bond breaking compared to broken-symmetry DFT. The CDFTCI error, however, approaches zero much slower than broken-symmetry DFT - we do
not understand this, but also do not consider this a serious problem as the absolute
magnitude of the error is small.
To judge the effect of dynamic correlation, we repeat the same calculation with
CASSCF(2,2) (Fig. 3-3). CASSCF(2,2) gives the right symmetry and energy at
dissociation, but badly underestimates the binding energy. We attribute this to the
lack of dynamic correlation at the equilibrium geometry. Near the equilibrium, at
0.75 Å, CASSCF(2,2) has an error of 14 kcal/mol while CDFT-CI B3LYP only has
87
20
Binding Energy (kcal/mol)
0
−20
−40
−60
−80
Full CI
CB3LYP−CI
RB3LYP
UB3LYP
CASSCF(2,2)
−100
−120
0.5
1
1.5
2
2.5
3
3.5
4
R (Å)
Figure 3-3: Figure taken from Ref. [1] Potential curves of H2 .
an error of 1 kcal/mol. This case demonstrates one of the motivating factors for
CDFT-CI - DFT’s ability to capture dynamic correlation.
3.4.3
LiF
The erroneous dissociation limit of highly ionic neutral molecules under DFT was
known early on [161], and has recently attracted more attention [162, 163]. You can
see this effect for the case of LiF in Fig. 3-5. Singly-bonded diatomic molecules
should always dissociate to neutral atoms, whereupon the singlet and triplet states
are degenerate. Hence, both singlet and triplet states of LiF should dissociate to
the same limit at infinity. Under unrestricted B3LYP, however, singlet LiF dissociates into atoms with erroneous fractional charges and a negative energy at infinity.
This is once again attributed to self-interaction error. Using the CAS(2,2) active
space, transformed to the localized basis Li↑F↓, Li↓F↑, Li+ F− and Li− F+ , the singlet
state under CDFT-CI once again dissociates correctly, while remaining accurate near
equilibrium.
Because conducting a Full-CI for this system size is impossible, we use an optimizedorbital coupled-cluster doubles [183, 184] calculation with the OD(2) second-order
correction in a CC-pVQZ basis for our comparison. The choice of unrestricted versus restricted Hartree-Fock as the reference state for the coupled-cluster calculation
88
Error of Binding Energy (kcal/mol)
5
CB3LYP−CI
UB3LYP
4
3
2
1
0
−1
−2
−3
0.5
1
1.5
2
2.5
3
3.5
4
R (Å)
Figure 3-4: Figure taken from Ref. [1] Error comparison of UB3LYP and CB3LYPCI. Errors are calculated against full CI results.
results in different energies [185]. We use the UHF reference, which gives the same
energy as RHF up to 5 Å. Observe in Fig. 3-6 that CDFT-CI gives a very good
equilibrium geometry and energy. At larger distances, however, the CDFT-MI results from BLYP and B3LYP differ. Assuming the OD(2) is a good proxy for full
CI, it would appear that CDFT-CI B3LYP is better than CDFT-CI BLYP. However,
overall both functionals perform pretty well.
We test sensitivity to the wc definition by trying another definition for the density
to atomic center mapping, the stockholder weight [186, 187]. The reference densities
used to obtain the stockholder weights are spherically-averaged atomic densities in
the 6-311G basis.
Ionic versus Covalent Components
With CDFT-CI, it is possible to identify the relative contributions of ionic and covalent terms in the ground state. We start by orthogonalizing the constraint operator
wc in the active space basis, much the same way it was previously done in the group
to obtain diabatic states for electron transfer [20]:
wc Ψi = ni SΨi
89
(3.4)
Binding Energy (kcal/mol)
50
0
−50
CB3LYP−CI
B3LYP (Ms=0)
B3LYP (Ms=1)
−100
−150
1
2
3
4
5
6
R (Å)
7
8
9
10
Figure 3-5: Figure taken from Ref. [1] Singlet potential curves of LiF by B3LYP and
CB3LYP-CI as compared to the triplet curve by B3LYP. The Ms =0 curve of B3LYP
is a broken-symmetry solution at large R and has a positive charge of 0.3 on Li at
the dissociation limit.
where the resulting states Ψi are the best orthogonal representations of the starting
active states, and have eigenvalues ni which represent the ionicity of the states. For
example, in the case of LiF, the constraint on Li has ni ≈ 2, 3, 3, 4. These values
correspond to the Li+ F− , LiF singlet/triplet, and Li− F+ states respectively. Taking
the overlap of the ground state with each basis state gives us coefficients, which
when squared sum to unity and individually represent the contribution from each
configuration. The weights as a function of bond distance are plotted in Fig. 3-7. As
expected, we see contributions from the ionic configuration Li+ F− and the covalent
states form most of the ground state at most distances. At short bond lengths,
the ionic state is dominant, while the covalent states are dominant at long bond
lengths. Configuration weights are smooth with respect to distance, and cross at
6.6 Å. A different way of estimating the crossing point is to ask when the difference
in ionic energies, as determined by the difference between IPLi and EAF , offsets the
electrostatic attraction between Li+ and F− . Under B3LYP, IPLi is 0.2064 Hartree
and EAF is 0.1270 Hartree, giving the crossing point at 6.6 Å, exactly where we found
it. This success is consistent with previous findings in the group that CDFT correctly
predicts long-range charge transfer energies.
90
Binding Energy (kcal/mol)
50
0
−50
OD(2)
CB3LYP−CI
CBLYP−CI
CB3LYP−CI (S)
−100
−150
1
2
3
4
5
6
R (Å)
7
8
9
10
Figure 3-6: Figure taken from Ref. [1] CB3LYP-CI(S), CB3LYP-CI and CBLYP-CI
potential curves as compared to OD(2). CB3LYP-CI(S) is the CB3LYP-CI using
stockholder weights.
3.5
Conclusion
We have introduced a configuration interaction method based on spatial and spin constrained states. The active space states in our method have local spatial and spin density constraints and treat dynamic correlation while remaining low in self-interaction
error and static correlation. Using this basis, we approximate the off-diagonal Hamiltonian matrix elements also using DFT, and at the end of the process add back static
correlation by diagonalization. This is efficient and provides a good treatment of both
dynamic and static correlation, enabling it to describe bond-breaking accurately. This
is evident from the good results we’ve obtained for the dissociation potentials of H+
2,
H2 and LiF.
There is a class of ab initio MR methods which perturb the ground state and use
the perturbed states as an active space for configuration interaction. CDFT-CI is
unique in this class because it is a purely DFT approach, using DFT to form both
the active space and compute matrix elements. Other “perturb-then-diagonalize”
approaches use perturbation theory, configuration interaction (in a bigger MO basis),
coupled cluster and canonical transformations [188, 189, 190, 191]. CDFT-CI is more
efficient than these techniques, and at the same time comparatively accurate for the
91
1
Weight
0.8
Li+F−
Neutral
Li−F+
0.6
0.4
0.2
0
1
2
3
4
5
6
R (Å)
7
8
9
10
Figure 3-7: Figure taken from Ref. [1] Weights of configurations in the final ground
state.
cases we have examined.
We attribute the success of the method to the nature of the active states chosen
- the active states we use have local spin and charge density, which minimizes selfinteraction error and static correlation. Avoiding the former controls an error endemic
to DFT, while having low levels of static correlation in the active space reduces doublecounting as static correlation is added later in the process. The dissociation curve
from the equilibrium bond length to dissociation is described correctly by CDFT-CI,
and this type of consistency is progress towards a reliable treatment of bond-breaking
and bond-formation in chemical reactions of larger molecules. Best of all, CDFTCI does this in a functional-agnostic manner, in theory with no greater reliance on
cancellation of error between SIE and static correlation biases. This will enable it to
take advantage of improvements in functionals which seek to reduce SIE [51, 52, 53].
3.6
Acknowledgements
This work was done in collaboration with Dr. Qin Wu, who is the first author on
Ref. [1].
92
Chapter 4
Rydberg Energies using Excited-State DFT
4.1
Introduction
As described in the thesis introduction, the Hohenberg-Kohn (HK) theorem guarantees that all the properties of the ground state can be extracted from its one-particle
density [13]. This existence proof has inspired the development of numerous functionals that approximate the ground state energy based on the density, and is the basis
of density functional theory (DFT).
There is, however, no corresponding theorem for electronic excited states. The
mapping from electronic excited states to densities is not unique [192] and thus a
one-to-one mapping does not exist. This means that it is not possible to construct a
functional which determines excited state energies from just their densities. However,
this hasn’t dissuaded the development of a number of ways of getting excited states
in the DFT framework. The most widely used is time dependent DFT (TDDFT),
where the Runge-Gross theorem [22] has proven the existence of a one-to-one mapping between the density trajectory and the time dependent potential. There are
also approaches which target excited states directly, including but not limited to
symmetry-based methods [193], methods based on ensembles [194] and methods that
attempt to treat individual excited states [142, 195].
Even though excited state DFT (eDFT) lacks a completely rigorous foundation,
it has been applied to a number of different systems of interest. A common approach
is to use the Kohn-Sham orbitals from a ground state DFT calculation, but occupy
93
them differently and compute the energy of the resulting density using a ground state
functional. This type of procedure has been used to treat excitons in conjugated
polymers [196, 197], exciton-induced bond breaking at semiconductor surfaces [198],
photoisomerization of aromatics [199], and solvatochromatic effects in dyes [200].
In this chapter, we attempt to do two things. Firstly, we try to apply the commonly used eDFT approach, which is usually limited to the treatment of HOMOLUMO transitions, to the description of higher-lying excited states. We use standard
semilocal functionals to treat atomic Rydberg states, and find that, even without
any correction of the exchange correlation potential, we are able to obtain reasonable
Rydberg excitation energies. We find this to be a result of, under eDFT, each Rydberg state experiencing its own effective potential - while each individual potential
decays exponentially beyond a certain distance, the distance at which the erroneous
exponential decay starts is pushed further and further out as one goes up the series
of Rydberg states.
Secondly, we attempt to reconcile the formally exact DFT-based excited state approaches mentioned above with the eDFT procedure. Specifically, we note that many
years ago, Levy and Perdew proved that all stationary densities of the exact functional, Ecs , (as defined by the Levy constrained search [201]) are physically significant
and in fact stationary states under the full Hamiltonian [202]. This theorem applies
to both minima (ground states) and saddle points (excited states) of the functional.
In this article, we explore the connection between these Levy-Perdew (LP) stationary
densities and the self-consistent solutions of the Kohn Sham (KS) eDFT equations.
We find that the two approaches are identical as long as the eDFT solutions give the
minimum KS kinetic energy. Thus, in certain situations one might conjecture that
eDFT excited states could be rigorous approximations to the true excited states. We
test this conjecture for the atomic Rydberg states computed herein and find that, in
some cases, the eDFT solutions are minima of the KS kinetic energy. In particular,
we find that that this occurs when the excited state density is not ground state vrepresentable. In these situations, there exists no potential for which the ground state
density is equal to the density in question. For these densities, then, there must be
94
an excited state solution which gives the minimum kinetic energy, and it is this state
that is rigorously predicted in eDFT. We thus find that eDFT plays a complementary
role to constrained DFT [203, 142, 1] in terms of describing excited states in DFT:
eDFT can only work if the density is not the ground state of some potential, while
constrained DFT can only work if it is.
4.2
Ground State Kohn-Sham DFT
We start by clarifying what we mean by eDFT, both to clarify notation and also to
specify the variant we employ (there are several in the literature). We start with
ground-state Kohn-Sham DFT (KS-DFT) [14]. Under KS-DFT, the energy
↑
↓
↑
↓
E[ρ , ρ ] ≡ Ts [ρ , ρ ] +
Z
ρ(r)Vext(r)dr + EJxc [ρ↑ , ρ↓ ]
(4.1)
is minimized with respect to the spin densities, which are expressed as a sum of KS
orbital densities
↓
↑
↑
ρ ≡
N
X
i
|φ↑i |
↓
ρ ≡
N
X
i
|φ↓i |
where ρ ≡ ρ↑ + ρ↓ and the KS kinetic energy is given by
↓
↑
↑
↓
Ts [ρ , ρ ] ≡ min
φ↑i →ρ↑
N
X
i
hφ↑i |T̂ |φ↑i i
+ min
φ↓i →ρ↓
N
X
i
hφ↓i |T̂ |φ↓i i
Physically, Ts is the minimum kinetic energy possible for a single-determinant
wavefunction that yields the given density, and the minimum is obtained for the
optimal KS orbitals φi . Meanwhile, EJxc contains all the inter-electron interactions
and must be approximated, in practice. Extremizing the energy expression (Eq. 4.1)
with respect to ρ↑ , ρ↓ subject to the normalization constraint gives
"
↑
↓
δ Ts [ρ , ρ ] +
Z
− µ↑
ρ(r)Vext (r)dr + EJxc [ρ↑ , ρ↓ ]
Z
ρ↑ (r)dr − N ↑
− µ↓
95
Z
ρ↓ (r)dr − N ↓
#
(4.2)
=0
The corresponding Euler equation implies that the Kohn-Sham orbitals φi↑,↓ satisfy
↑
T̂s + V̂ext + V̂Jxc
[ρ↑ , ρ↓ ] · φ↑i = ǫ↑i · φ↑i
↓
↑ ↓
T̂s + V̂ext + V̂Jxc [ρ , ρ ] · φ↓i = ǫ↓i · φ↓i
(4.3)
Solving the Kohn-Sham equations(Eqs. 4.3) requires a set of self consistent iterations,
because VJxc depends on the density while the density depends on VJxc via the orbitals. For grounds states, this leads to the Kohn-Sham self-consistent field (KS-SCF)
iterations:
↑,↓
1. Compute the potentials VJxc
from the density ρ↑,↓ .
2. Determine the orbitals that solve the KS equations (Eqs. 4.3) for the fixed
↑,↓
potentials VJxc
.
3. Choose the N ↑ (N ↓ ) lowest spin-↑ (spin-↓) orbitals (Aufbau occupation), and
form new spin densities ρ↑new , ρ↓new from these orbitals.
4. If ρnew is sufficiently similar to ρ, the procedure is done. Otherwise, start over
from step 1 with ρnew .
Due to Janak’s theorem [204], the Aufbau occupation in step 3 implies that the
converged KS-SCF solutions are energy minima - that is to say, they correspond to
ground states.
4.3
Excited State Kohn-Sham DFT
In excited state DFT, we follow the KS prescription in all respects, except that we
modify step 3 to allow non-Aufbau occupations. All calculations are still to be per↑,↓ ↑ ↓
formed self-consistently, so that the potentials VJxc
[ρ , ρ ] will be different for different
excited states, because they will be evaluated using different excited state densities.
Of the many non-Aufbau schemes possible, the simplest consists of occupying the
orbitals as if the Hamiltonian were non-interacting, e.g. the first excited state would
fill the N-1 lowest orbitals and the (N+1)-th lowest orbital, swapping the HOMO and
96
LUMO occupations as compared to ground state, while other excites states would correspond to HOMO ↔ LUMO+1, HOMO-1 ↔ LUMO,.... This scheme often works
well for the lowest excited state, particularly if the HOMO and LUMO are of different
symmetries [196, 197, 198, 199, 200]. However, for higher-lying states we find this
simple approach rarely converges to a self-consistent solution. Changes in orbital energy levels between iterations often permute the order of the physical orbitals, leading
what was, say, the LUMO+2 in the previous iteration to be LUMO+1. These erratic orbital ordering swaps lead to inconsistent orbital occupations and the resulting
changes in the density from iteration to iteration then lead to non-convergence in
most cases.
Explicitly designing for convergence, we devised a scheme that minimizes density
difference relative to a reference density. The basic idea is that, within the selfconsistent iterations one occupies the KS orbitals that have greatest overlap with a
set of reference orbitals. Since we do not use energetic ordering, this removes the
problem of erratic orbital occupation. In order to obtain good reference densities,
we actually use an adiabatic connection between the non-interacting and interacting
systems. Here, we reduce the electron-electron interaction terms by a factor λ and
begin with the non-interacting limit (λ = 0) where exact excited states can be trivially
obtained for any non-Aufbau configuration. We then use these orbitals as a starting
point for a calculation at λ = ǫ (where ǫ << 1). Once this calculation is converged,
we use the λ = ǫ orbitals as a reference for a self-consistent λ = 2ǫ calculation,....
The solution is thus brought adiabatically from the non-interacting (λ = 0) to the
fully-interacting (λ = 1) limit. This method is well-defined but probably sub-optimal
- we leave the search for efficiency to future work.
4.4
State of the Art: Linear Response TDDFT
A cheap and accurate DFT-based excited-state method already exists in the form of
time-dependent DFT (TDDFT) and thus an approach like eDFT should be thought
of as a supplement to TDDFT, filling in the gaps where TDDFT is to cumbersome or
existing functionals are not accurate enough. In particular, we note that TDDFT has
97
severe problems treating Rydberg excitations accurately [205, 206] and the reasons for
this are well understood. Specifically, whereas the true exchange-correlation potential,
Vxc should fall off as − 1r at large distances from a molecule, the potential predicted
with commonly used approximate functionals decays exponentially [207, 4]. Further,
the LUMO in Kohn-Sham DFT systematically underestimates the electron affinity
due to the lack of a derivative discontinuity, which is known to be present in the exact
functional [40, 41, 42]. For localized, valence-like excitations, these are not serious
problems. However, because most of the electron density for Rydberg states resides
in the asymptotic region, they are extremely sensitive to the long range behavior
of Vxc , leading to a systematic overestimate of Rydberg transitions. In the worst
cases, the Rydberg states are so overestimated that they lie above the ionization
threshold and are thus embedded the continuum. There are a number of ways to
solve this problem within TDDFT. On the pragmatic side, one can simply replace
the approximate Vxc with the correct − 1r + ∆E decay after some appropriate distance
[207, 4]. The asymptotic form will have little influence on the valence excited states,
but dramatically improves the Rydberg states. On the more formal side, it has
recently been shown that many problems with Rydberg states in TDDFT can be
circumvented via clever use of quantum defect theory [208]. Here we explore the idea
that eDFT could provide a third route to Rydberg states in DFT.
4.5
4.5.1
Application
The Excited-State Exchange-Correlation Potential
To start with, we examine the simplest Rydberg states; the s-type eigenstates of the
hydrogen atom. To further simplify matters, we will also use the known exact densities
for the hydrogen Rydberg states our analysis, postponing all questions concerning selfconsistency to the next section. Now, we want compare TDDFT and eDFT using the
same functional in each case. The only difference is that in eDFT the functional is
evaluated at the excited-state density, whereas the TDDFT potential is determined
by the ground state. This leads to a different exchange-correlation potential in each
case and – as we shall see – vastly different behavior for Rydberg states.
98
Electronic State
1s
2s
2p
3s
3p
3d
ǫ1s
-6.32
-12.00
-10.62
-13.58
-13.10
-11.32
ǫ2s
0.91
-1.80
-1.30
-2.43
-2.12
-2.20
ǫ2p
ǫ3s
ǫ3p
ǫ3d
1.41 0.82 1.02 1.33
-1.41 0.03 0.15 0.10
-1.28 0.19 0.23 0.32
-2.41 -0.84 -0.76 -0.55
-2.34 -0.73 -0.76 -0.52
-2.08 -0.51 -0.49 -0.53
Table 4.1: eDFT orbital eigenvalues of the hydrogen atom computed using LSDA
exchange and the exact densities. Each row corresponds to a particular excited state,
and the columns give the appropriate orbital eigenvalues. Note that, as one occupies
more and more diffuse orbitals, the 1s orbital eigenvalue approaches the correct value
of -13.6 eV, naturally correcting for the poor LSDA exchange-only estimate of the
ionization energy of ǫ1s =-6.3 eV.
As is well known, the failure of density functionals for Rydberg states stems from
two qualitative problems: 1) the apparent ionization threshold of the atom or molecule
must be adjusted [207, 4] to account for the derivative discontinuity of the xc-potential
[40, 41, 42] and 2) the xc-potential should decay as 1/r at large distances, while commonly used functionals typically decay exponentially [209]. We first show analytically
that the prescription of eDFT addresses both of these concerns without modifying the
xc functional. To address the first point, Table 4.1 shows the computed Kohn-Sham
eigenvalues (ǫ) for the 1s, 2s, 2p, 3s ... orbitals when the electron is considered as
occupying any one of the 1s, 2s, 2p, 3s... states. If we neglect self consistency considerations, then, the first row represents the ground state orbital eigenvalues, while
subsequent rows represent the first, second, third... excited state eigenvalues as would
be obtained from eDFT. Examining the data in the table, it is clear that the ground
state HOMO underpredicts the ionization potential (IP) for hydrogen (e.g ǫHOM O =6.3 eV which should be -13.6 eV). The result is that any Rydberg excitation that
exceeds 6.3 eV will be embedded in a continuum; indeed TD-LSDA exchange predicts no bound excited states for hydrogen. However, looking at the subsequent
columns in the table, we see that as one moves up from first, to second, to third
... excited states, the orbital energies shift so that the 1s orbital quickly approaches
the correct IP. Note for example, that when the electron sits in the 3s orbital, the
eigenvalue of the 1s orbital is -13.58 eV. Thus, the apparent ionization threshold in
99
eDFT is not related to the ground state orbital eigenvalues. We conclude that the
eDFT prescription incorporates some of the effects of the derivative discontinuity on
the orbital eigenvalues, pushing the IP high enough that Rydberg-like states should
be bound.
0
Energy/eV
-5
-10
-15
-20
Exact
LSDA 1s
LSDA 2s
LSDA 3s
-25
-30
0
2
4
6 8 10 12 14 16 18 20
Radius/Angstroms
Figure 4-1: Hydrogen atom excited state exchange potentials (eV vs. Angstroms).
The solid red line shows the exact exchange potential and the subsequent lines (moving upward at the origin) show the 1s, 2s and 3s LSDA exchange potentials from
eDFT using the exact density. While none of the potentials individually has the
correct decay, subsequent potentials decay more and more slowly.
Addressing the potentials, Figure 4-1 shows the LSDA exchange potentials that
result from the 1s, 2s and 3s hydrogen orbitals. We present only the exchange potential for simplicity, since it is exchange (and not correlation) that determines the
long-range behavior of Vxc Clearly, none of the potentials is particularly similar to
the exact potential. They all decay too quickly in the asymptotic region, and once
we get beyond the ground state, the potential even has nodes at finite values of r,
corresponding to the nodes in the hydrogenic orbitals. However, hidden within this
series of potentials is the information necessary for a Rydberg-like series, as we now
show.
We focus on the region near the classical turning point (rtp (n) = 2n2 ), where the
density will tend to be large. If we examine the large-n limit appropriate for Rydberg
100
states, we can use Erdélyi’s asymptotic expansion of the Laguerre polynomials to
facilitate simple analytic expressions for the exact density and density gradient [210].
We then get the following expressions for the exact hydrogen s-orbital density and
gradient at the classical turning point rtp (n) = 2n2 :
lim ρn (r)|r=2n2 ≈
n→∞
n−17/3
√
22/3 27 3 3πΓ
=
5 2
3
0.00632
n17/3
(4.4)
n−7
−0.09188
lim ∂r ρn (r)|r=2n2 ≈ − √ =
n→∞
n7
2 3π
q
3
lim ∂r2 ρn (r)|r=2n2 ≈
n→∞
3
Γ
2
2 2
3
16π 3
n−25/3 =
0.00423
n25/3
Again, focusing on exchange, we take LSDA exchange
1/3
6
vLSDA (ρ) = −
ρ1/3
π
(4.5)
and equation 4.4, the LSDA exchange potential vLSDA at the classical turning point
is
lim vLSDA (ρn (r)|r=2n2 ) ≈
n→∞
√
−92
37/9 n17/9 πΓ
5
3
2/3 =
−0.229
n17/9
(4.6)
The dimensionless density gradient, s = |∇ρ|/(2kf ρ), where ρ = kF3 /3π 2 , as a
function of the principle quantum number n and evaluated at rtp is
lim s(r)|r=2n2
n→∞
8/3
317/18 Γ 23
√
n5/9 ≈ 0.4351n5/9
=
9
5/3
2 2π
(4.7)
Under the same conditions, the PBE exchange functional [30] is
lim vP BE (ρ, |∇ρ|) = −
n→∞
0.414
n17/9
(4.8)
and the PW91 exchange functional [29] is
lim vP W 91 (ρ, |∇ρ|) = Fx (ρ, |∇ρ|)∂ρ vlda (ρ) + ∂ρ Fx (ρ, |∇ρ|)vlda (ρ) = −
n→∞
101
0.206
n5/3
Under the same conditions, the Becke88 exchange functional [31] gives
lim vBecke88 (ρ, |∇ρ|) =
n→∞
0.312
− log
−
2
(n)
n4/3
1.87
log3 (n)
+
−0.229 +
5.28
log2 (n)
n17/9
+
20.2
log3 (n)
+O
1
n2
0.1
Energy/eV
0
-0.1
-0.2
-0.3
-0.4
Exact
LSDA 1s
turning point LSDA ns
-0.5
0
500 1000 1500 2000 2500 3000 3500
Radius/Angstroms
Figure 4-2: Exchange potentials for hydrogen. The dotted green line is the groundstate LSDA exchange potential [2]. The solid red line is the exact exchange potential.
The blue crosses are the excited-state density LSDA exchange potential at the classical
turning points r = 2n2 .
Comparing the approximate potentials to the exact exchange potential at the
turning point,
vx (r)|r=2n2 = −
0.5
n2
(4.9)
we see that both the exact and the excited-state LSDA, Becke88, PBE and PW91
exchange potentials are polynomials in n. In contrast, the LSDA exchange potential
of the exact ground-state density is
1/3
3
2
2
e−4n /3 = −1.97e−4n /3
vLSDA (ρ1 (r)|r=2n2 ) = −2
π
102
As evident from figure 4-2, the ground-state LSDA exchange potential bears little resemblance to the exact exchange potential, whereas the excited-state LSDA exchange
is close to the exact value over a large range in n: the effective potential in eDFT
shows a marked improvement over the ground state potential, and should be expected
to give a more faithful description of Rydberg states. To understand why this is the
case, note that within the adiabatic local density approximation, Vxc decays exponentially as a consequence of two factors: the exponential decay of the density and
the local nature of the functional. In the the large r limit, any given excited-state Vxc
also decays exponentially, for precisely the same reasons (cf Fig. 4-1). However, as n
increases, the region of exponential decay for the excited state Vxc gets further and
further from the nucleus. Thus, because each excited state sees its own xc potential,
areas of significant density need not coincide with the asymptotic r region. Indeed,
as shown above the regions of large density near the classical turning points experience an effective polynomial decay with distance; thus, for a given excited-state,
the electron density never “sees” the exponential decay of the excited-state exchange
potential.
4.5.2
nl
2s
2p
3s
3p
3d
4s
4p
4d
4f
5s
Avg
RMS
Numerical Evaluation of Rydberg Energies
Eexact
10.204
10.204
12.094
12.094
12.094
12.755
12.755
12.755
12.755
13.062
ELDA−X
9.222 (-0.982)
9.195 (-1.009)
10.981 (-1.113)
10.909 (-1.185)
10.912 (-1.182)
11.608 (-1.147)
11.563 (-1.192)
11.533 (-1.222)
11.534 (-1.221)
11.902 (-1.159)
-1.141
1.144
ELSDA
9.502 (-0.702)
9.423 (-0.782)
11.374 (-0.720)
11.261 (-0.833)
11.247 (-0.847)
12.062 (-0.693)
11.948 (-0.808)
11.947 (-0.808)
11.930 (-0.826)
12.391 (-0.670)
-0.769
0.771
EP W 91
9.955 (-0.250)
9.830 (-0.374)
11.910 (-0.185)
11.776 (-0.318)
11.745 (-0.348)
12.635 (-0.120)
12.496 (-0.260)
12.494 (-0.262)
12.469 (-0.287)
12.979 (-0.082)
-0.248
0.264
EB3LY P
9.985 (-0.219)
9.874 (-0.330)
12.625 ( 0.531)
11.806 (-0.288)
11.767 (-0.327)
NC
NC
NC
NC
NC
-0.127
0.355
Table 4.2: Hydrogen Atom eDFT Excitation Energies (eV). Italics indicate difference
from the exact energy. NC indicates no converged state was found. Average and Root
Mean Square errors are also indicated.
So far, we have only examined the exchange-correlation potentials as calculated
103
from frozen densities. We test eDFT by self-consistently calculating the energies of
atomic Rydberg states using the techniques described above. To begin with, we have
computed the eDFT excitation energies of the hydrogen atom using several commonly
used functionals [29, 136], with the excitation energy being given by the difference
between self-consistent excited-state and ground-state energies. In order to obtain
complete basis set-quality results, we utilize an uncontracted d-aug-cc-pV6Z basis,
consisting of 100 uncontracted Gaussians [211], leading to absolute energies that are
converged to within 0.05 eV in all cases. Our results are presented in Table 4.2.
Examining the results, we see that each functional does, in fact yield a Rydberg-like
series of self-consistent excited states. This is perhaps to be expected from the analysis
of the potentials detailed in the previous section, but it is nonetheless comforting
that self-consistency does nothing to spoil this property. We also note that, while
the previous section focused solely on s-states for simplicity, our calculations reveal
the presence of s-, p-, d- and f-type Rydberg states with similar accuracy. There is a
systematic trend in these eDFT results toward underestimating the Rydberg energies
by several tenths of an eV. This observation should be contrasted, however, with the
fact that TDDFT with the same functionals predicts no bound Rydberg states, with
all the Rydberg-like solutions being embedded in the continuum. Hence, from this
perspective, eDFT provides a dramatic improvement to the TDDFT description of
these states. On the other hand, these results could be seen as disappointing, since
even something as simple as Hartree-Fock predicts the Rydberg series of hydrogen
exactly.
In any case, we do see the expected Jacob’s Ladder [212] trend in accuracy
(Local<GGA<Hybrid), which suggests semi-systematic approach to the correct answer. On the other hand, we find that the ease of convergence is inversely related
to accuracy: the LDA exchange only calculations are the easiest to converge (in that
the adiabatic connection can be achieved with large steps in λ) while it was not even
possible to converge some of the Rydberg states for B3LYP. The primary obstacle to
convergence is that the high-lying states have a tendency to collapse to lower-lying
solutions even when the connection is performed very slowly. We finally note that the
104
degeneracy of different l values for a given n is very nearly preserved by the eDFT
predictions. Since this is a special property of the Coulomb potential, one cannot
expect perfect fidelity from the approximate functionals, and the clustering of eDFT
results into quasidegenerate groups is a relatively positive result.
nl
2p
3s
3p
EHF
1.842 ( 0.020)
3.335 ( 0.010)
3.798 ( 0.018)
ELDA−X
1.576 (-0.246)
3.018 (-0.308)
3.304 (-0.477)
EB3LY P
1.607 (-0.215)
3.278 (-0.048)
3.589 (-0.191)
ET DLDA
1.972 ( 0.150)
3.078 (-0.247)
NA
ECS00
1.971 ( 0.149)
3.151 (-0.175)
3.410 (-0.371)
EExpt.
1.822
3.326
3.780
3d
3.831 ( 0.007)
3.502 (-0.322)
3.824
0.013
0.014
3.682 (-0.142)
3.583 (-0.241)
-0.167
0.180
NA
avg
rms
3.406 (-0.418)
3.161 (-0.663)
-0.422
0.446
-0.049
0.205
-0.180
0.271
Table 4.3: Lithium Atom Excitation Energies (eV). Italics indicate difference from
the experimental energy. ELDA−X and EB3LY P denote eDFT energies with the appropriate functional, while TDLDA and TDB3LYP are the corresponding TDDFT
energies. CS00 [4] contains an asymptotic corrections to the B3LYP exchange potential. In the TDDFT columns NA indicates energy levels above the ionization energy.
Experimental numbers from Ref. [5].
As a somewhat more complex test of our algorithm, we have examined the excited
states of the Lithium atom. Our calculations employed an uncontracted aug-ccpVQZ basis [211], which led to excitation energies correct to within 0.1 eV. Our
results are presented in Table 4.3. Clearly, once again, eDFT gives relatively accurate
Rydberg energies using standard functionals, even for states that TDDFT predicts are
embedded in the continuum. Indeed, for B3LYP, the Rydberg energies are predicted
with an accuracy approaching that achieved for valence states (e.g. within .2 eV). The
eDFT predictions once again systematically underestimate the experimental results.
Further, we note that, in our simulations the eDFT d-states typically broke symmetry.
This splitting arises because the eDFT exchange-correlation potential is fully statespecific – each of the five d orbitals experiences its own (non-spherical) potential.
It is thus possible these results could be improved by using a functional that treats
degenerate members of a single term on an equal footing[213].
Hartree Fock gives an extremely accurate treatment of Rydberg states generally,
and therefore it is not surprising that HF soundly outperforms eDFT for these ex105
citation energies. In order to get a ballpark feeling for the accuracy of these eDFT
energies, we also compare them to asymptotically corrected (AC) Casida-Salahub
(CS00) [4] TDDFT results. CS00 creates a clear Rydberg series, showing that there
are bound 3p- and 3d- states for the lithium atom. However, the results are not
significantly better than the eDFT predictions, although CS00 does preserve the degeneracy of the d-levels. While this may or may not be indicative of the quality
of eDFT for more complicated Rydberg states, it suggests that eDFT might be a
pragmatic means of obtaining Rydberg excitation energies in DFT.
Before moving on, we should offer the caveat that, even with the adiabatic connection algorithm outlined above, obtaining converged eDFT excited states for Li
was extremely difficult. Some part of this difficulty was due to the very diffuse basis
set required to obtain accurate energies. However, a large portion also arose from
the inherent instability of solving the eDFT equations. Thus, while one would ideally
like Rydberg excited states for several more atoms to make a quantitative comparison
between AC functionals and eDFT, such a comparison is impractical with the present
algorithm.
4.6
4.6.1
Discussion
Extrema of the ground state energy functional
Are the results above merely fortuitous, or is there some deeper reason why one should
expect eDFT to provide good excitation energies from a delta SCF procedure? It is
known that there is no universal functional that gives all excited state energies, but
is it possible that there exists a universal functional that gives exact energies for a
subset of excited state densities? We suspect that the answer to the latter question is
’yes’ and based on a theorem proven by Perdew and Levy concerning extrema of the
ground-state energy functional [202]. We review the theorem here for completeness
and then proceed to explore its applicability to the present results.
Begin with a constrained search formulation of the ground state energy functional
106
Ev [ρ]
Ev [ρ] ≡ min hΨ|Ĥv |Ψi
Ψ→ρ
The absolute minimum of this functional is the ground state energy, E0 , and it
assumes this value at the ground state density ρ0 :
E0 = Ev [ρ0 ] = min Ev [ρ].
ρ
Perdew and Levy proved that every extreme density,ρi (maximum, minimum, saddle
point) of Ev is the density of a stationary state of Ĥv and the value of Ev at that
density is the lowest stationary state energy of that density. Thus, while the minimum
of Ev has a rigorous interpretation in connection with the ground state, all other
extrema of Ev can just as rigorously be associated with excited states. Further,
Perdew and Levy were able to show that no extremal density of Ev above the ground
state could be pure-state v-representable – that is, the excited state densities found by
this procedure must not be ground state densities for any other external potential [214,
215, 216]. Thus, it may be possible to use the ground state functional to determine the
energies of certain excited states, so long as the associated excited state density is not
the ground state density of some other potential. Further, it would seem likely that
highly excited Rydberg densities would be likely candidates if one is searching for a
non-pure state v-representable density. Hence, the Perdew-Levy theorem would seem
to suggest (but not prove) that Rydberg excited states energies might be rigorously
obtained from appropriate extrema of the ground state functional Ev
In order to flesh out the connection of the above theorem with eDFT, we employ
a constrained-search formulation of KS theory [217]. Here, one defines the KS kinetic
energy functional via:
Ts [ρ] ≡ min hΨ|T̂ |Ψi
Ψ→ρ
107
(4.10)
and the Coulomb-exchange-correlation energy functional via
Exc ≡ Ev [ρ] − Ts [ρ] −
Z
ρ(r)v(r)dr.
The Euler equation for an extremum of Ev being given by
δEv
=µ
δρ
one finds that, for the ith extremal density,
δTs + vs [ρi ](r) = µ
δρ(r) ρi
(4.11)
Eq. 4.11 is precisely the Euler equation one would obtain for a system of noninteracting electrons moving in a potential vs – this is the Kohn-Sham reference
potential associated with the external potential v. We further note that, Eq. 4.11
holds independent of whether one is speaking of a minimum, maximum or saddle
point of Ev . Thus, for every stationary density, ρi of Ev , there exists a noninteracting
system that reproduces the exact density, and that system moves under the associated Kohn-Sham potential vs [ρi ]. Thus, there is a close connection between eDFT
excited states – which by design are self-consistent solutions of Eq. 4.11 (cf Eq. 4.3 –
and the densities that extremize Ev .
Now, Eq. 4.11 is a necessary but not sufficient condition for a given density to
make Ev stationary. In order to satisfy Eq. 4.10, one must additionally have that the
Kohn-Sham reference state minimizes the kinetic energy for the given density. The
question therefore arises: do the Rydberg states determined in the previous section
deliver the minimum kinetic energy for their density? If they do, then the associated
eDFT energies are rigorous approximations to the correct Rydberg energies, within
the approximations due to functional, basis set, etc. If there is a non-interacting
reference with the same density but lower energy, then the states are likely artifacts
and the agreement with true Rydberg results is likely fortuitous.
We have attempted to address this question for the Rydberg states above. For
108
the hydrogen atom, it is trivial to show that every eDFT solution delivers the lowest
kinetic energy for its density. Since there is only one electron, there is only ever one
wavefunction that maps to a given density (up to an overall phase) and thus there
can only be one non-interacting system associated with each excited state density.
Thus, the hydrogen atom excited state energies predicted above should be considered
rigorous approximations.
The situation for lithium is more complicated, as for this three electron system
there are always an infinite number of non-interacting states that map to a given
density. In order to locate the minimum kinetic energy associated with each lithium
density, we employed a direct minimization technique [218] that is essentially a refined
version of the Zhao-Morrison-Parr (ZMP) procedure[219]. Specifically, for a given
electron density ρin , one defines the following new functional of any determinantal
wave function Ψdet and the Lagrange multiplier function v(r)
Ws [Ψdet , v(r)] =
N
X
i
hφi |T̂ |φi i +
Z
drv(r) {ρ(r) − ρin (r)} ,
(4.12)
where φi are the orbitals and v(r) can be regarded as the corresponding potential,
which is expanded in a basis [218], with the expansion coefficients treated as variational parameters. The solution is found by optimizing Ws . When Ws is optimized,
P
the difference between Ws and Ts (= N
i hφi |T̂ |φi i) shows how well the given density
is reproduced. We note that the v-representability question arises here, as well. The
above formalism is generally applicable any density, either from the ground state or
excited states. Meanwhile, when building Ψdet , one is not required to always use
orbitals with lowest energies, i.e, one may leave a hole below the highest occupied
orbital. In practice, we always stay in the Kohn-Sham scheme; that is, orbitals with
lower energies are always occupied first. Hence, if the optimized potential does not
reproduce Ws , ρin , then ρin is probably not noninteracting v-representable. With
this caveat, we can state the results of our search quite succinctly: for all the s-type
excited states in Table 4.3, we were able to find an alternate non-interacting ground
state that gave a lower KS kinetic energy. For the p- and d- type states our ZMP
109
search was unsuccessful in locating a ground state that reproduces ρin . This result
does not prove conclusively that the p- and d- densities are not noninteracting vrepresentable, as our search is only numerical and does not necessarily exhaustively
cover the available density space. However, the results strongly suggest that the pand d- densities are not noninteracting v-representable.
We thus conclude that the s-type excited states for lithium are accurate, at least
in part, due to a cancellation of errors, because their kinetic energies are not what
would be predicted by the constrained search. A cancellation of errors could be
expected, however, as Rydberg states are essentially one-electron in nature, making
the KS reference a fairly good description of these excitations. On the other hand,
our results suggest that the p- and d-type Rydberg states may be more physically
justified, since their densities appear to be not noninteracting v-representable. This
may stem, in part, from the symmetry considerations present in the p- and d- excited
states.
Given the intimate connection between the physical basis of eDFT and the pure
state v-representability of the density, one immediately recognizes a connection between eDFT and constrained density functional theory [203]. In constrained DFT,
one applies a constraint to the electron density in order to enforce some physical property of interest (e.g. dipole, charge localization) 1 . As we have recently shown, this
technique can be used to good effect to obtain information about charge transfer excited states [142, 19, 21, 220] and spin states [221, 143]. In each case, the excited state
in question can actually be obtained as the ground state of the system under an alternative potential. Taken together with our results here, one comes to the conclusion
that these techniques are essentially complementary to one another; while constrained
DFT works only when the excited density is pure state v-representable, eDFT appears to only apply when the excited density is not pure state v-representable. Thus,
while neither of these techniques by itself provides complete coverage of the space of
1
We note that, somewhat confusingly, a few authors have historically referred to what we here
call excited state DFT, as constrained DFT, with the idea that eDFT involves a constraint on the
occupation numbers. For our purposes, constrained DFT will refer only to constraints involving the
density.
110
all possible excited states available in DFT, together they bring us a few steps closer
to a rigorous basis for obtaining excited states as stationary states in DFT.
4.7
Conclusions
We have investigated both the practical and formal utility of excited state DFT
(eDFT) calculations for electronic excited states, with particular emphasis on Rydberg states. Because the Kohn-Sham (KS) potential in eDFT is different for each
excited state we find that eDFT gives a qualitatively correct asymptotic form of the
effective potential, even for commonly used exchange correlation functionals for which
the ground state potential has the incorrect form at large distances. We verify numerically that commonly used functionals – including LSDA, GGAs and hybrids – give
accurate energies for the Rydberg states of the hydrogen and lithium atoms. While
the errors in these cases are worse than Hartree-Fock, they are competitive with at
least one asymptotically corrected hybrid functional [4]. Finally, we address the formal basis of eDFT for excited states. Based on a theorem due to Perdew and Levy
[202] we note that all stationary energies of the exact functional correspond to eigenvalues of the true Hamiltonian. We further note that eDFT solutions correspond to
stationary states as long as they deliver the minimum kinetic energy for their density.
We test this criterion in the present cases, and find that for all the hydrogen states
and all of the p- and d- states for lithium, the eDFT solution delivers the minimum
kinetic energy for its density. Thus, in some cases, eDFT can be a rigorous way to
obtain excited state energies in DFT.
Moving forward, there are several issues to be addressed. The primary obstacle to
progress relates to convergence; even for these simple systems, we found it extremely
difficult to converge the self-consistent field (SCF) iterations for eDFT, as high-lying
solutions had a very strong tendency to collapse to lower-lying states. In order for
eDFT to become in any way practical, then, one must determine a means of stabilizing
the SCF iterations, perhaps using variations of existing SCF minimization strategies
[8, 9, 222] adapted to locate saddle points. The second major issue is the necessity
of testing to see if the converged eDFT solution delivers the minimum kinetic energy.
111
While not computationally costly, this procedure is far from standard and the option
to minimize Ts is not available in most production codes. Do eDFT states that are
not minima give appreciably worse results than those that do? Can one build in this
constraint to the SCF iterations?
Finally, on the more speculative side, there are a number of directions one would
like to take this result assuming the technical obstacles mentioned above can be
overcome. The Perdew-Levy theorem suggests that eDFT provides a route to excited
states whose densities are not the ground state of some other potential (i.e. they
are not pure-state v-representable). This puts eDFT in a complementary position to
constrained DFT, which is explicitly able to approximate excited states that are the
ground state of an alternative potential [203]. Thus, while neither of these techniques
together provides a complete set of excited states (as can be obtained in principle from
TDDFT) together these two methods may cover a large fraction of the “interesting”
low-lying excited states of atoms and molecules. For example, we have shown that
constrained DFT provides a rigorous route to charge-transfer excited states [142, 19]
while eDFT describes Rydberg states effectively. It would be interesting to see if
some combination of these techniques could be used to obtain doubly excited states
with DFT, which have proven elusive in TDDFT [223].
112
Chapter 5
Conclusion
Even though DFT is formally exact, existing approximations to the DFT energy
functional suffer from numerous flaws. These flaws are broadly attributed to selfinteraction error, the adiabatic approximation, and spatial locality of xc functionals.
The improvement of exchange-correlation functionals is an active field of research,
and much of that work is geared towards ridding exchange-correlation functionals
of the artifacts described above. It was realized pretty early on, however, that the
success of DFT was in no small part due to the fortuitous cancellation of error between
the different artifacts. As such, the independent elimination of single artifacts from
the list often led to worse results in practice.
Our answer to this conundrum has been to tackle the xc functional artifacts indirectly - instead of developing functionals, we have focused on developing procedures
which overcome some of the systematic flaws, but which would also work with an exact functional. It is our hope that in doing so, we are making progress in a direction
orthogonal and complementary to that of functional developers.
Practically, this orthogonality facilitates the application of existing functionals to
new problems. Intellectually, however, the successes and failures of our approaches
can help to guide functional improvement. Wavefunction-based methods may seem
to provide the ideal reference for DFT development by eliminating the uncertainty
associated with the interpretation of experimental results. However, systems of interest to DFT practitioners are often too big for high quality wavefunction methods
to be used. As such, alternative protocols for determining physical observables, like
113
the procedures described in this work, make available self-consistency checks and in
doing so provide clues as to the nature of inaccuracies encountered.
114
Appendix A
Magnus and Runge-Kutta Munthe-Kaas Integrators
Both of these integrators are strictly unitary, with unitarity ensured by the use of
numerical diagonalization routines.
A.1
Magnus Integrators
The 2nd order propagator is presented in Chapter 3. Here we present the 4th order
Magnus integrator scheme.
Û (t, t + dt) ≈ exp(Ω̂1 + Ω̂2 ) + O(dt5 )
dt
Ω̂1 = −i · (Ĥ(τ1 ) + Ĥ(τ2 )) ·
2
√
2
3 dt
Ω̂2 = [Ĥ(τ1 ), Ĥ(τ2 )] ·
12
√
3
1
τ1,2 = t + ( ∓
) · dt
2
6
Like the 2nd order propagator, because of the implicit time-dependence of the
Fock matrix we need to formulate a predictor-corrector scheme, which is depicted in
figure A-1.
1. Polynomial extrapolation: 3rd order fock matrices stored from previous time
steps, 1a, 1b and 1c, are used to extrapolate the fock matrices 3a and 3b.
115
Figure A-1: Extrapolation routine for the 4th order Magnus integrator. The order
row shows the time order (in dt) to which the matrices in the same column are correct
to.
2. Magnus Integrator: using 3a and 3b, the density matrix 2 is propagated to 4.
3. Fockbuild: density matrix 4 is used to compute the fock matrix 5.
4. Polynomial extrapolation: fock matrices 1b, 1c and 5 are used to extrapolate
fock matrices 6a and 6b.
5. Magnus Integrator: fock matrices 6a and 6b are used to propagate the density
matrix 4 to density matrix 7.
6. Fockbuild: density matrix 7 is used to compute the fock matrix 8.
7. Magnus Integrator: fock matrices 5 and 8 are then used to propagate the density
matrix 2 to density matrix 9.
A.2
RKMK Integrators
By using the ansatz
Ψ(t) = eΩ̂(t) · Ψ(t0 )
one can reformulate the schroedinger equation as a differential equation for Ω̂(t),
and then propagate Ω̂(t) using Runge-Kutta. This is known as the RKMK [224]
116
propagator. The RKMK propagator uses the same number of fockbuilds as regular
Runge-Kutta, but is unitary.
∂t eΩ̂(t) = dexpΩ̂(t) ∂t Ω̂(t) · eΩ(t)
dexp =
exp(ad ) − I
adÂ
ad (B̂) = [Â, B̂]
∂t eΩ(t) · Ψ(t0 ) = dexpΩ(t) ∂t Ω̂(t) · eΩ(t) · Ψ(t0 )
∂t Ψ(t) = dexpΩ(t) ∂t Ω̂(t) · Ψ(t)
−iF̂ (t) · Ψ(t) = dexpΩ(t) ∂t Ω̂(t) · Ψ(t)
−i
F̂
(t)
= ∂t Ω̂(t)
dexp−1
Ω̂(t)
adÂ
exp(ad ) − I
∞
X
Bj j
ad
=
j! Â
j=0
dexp−1
=
Â
where Bj are the Bernoulli numbers.
117
118
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134
Chiao-Lun Cheng
11 Tg Rhu Rd # 18-01
Singapore 436896
http://www.thechiao.com
+65 6345-0355
chiao@alum.mit.edu
Objective
• Computational Physicist looking to solve challenging complex problems with
strong analytical skills and systematic, mathematical mindset
Education
Massachusetts Institute of Technology
• Ph.D., Physical Chemistry, GPA: 5.000/5.000, 2008
GRE Verbal: 630/800, Math: 800/800, Writing: 4.5/6
Cambridge, MA
University of California, Berkeley
Berkeley, CA
B.S. Chemistry, High Honors, GPA: 3.844/4.000, 2003
•
B.A. Molecular and Cell Biology, High Distinction in General Scholarship, 2003
Physics Minor, 2003
•
University of California, Irvine
Chemistry and Biology Double Major, GPA: 3.886/4.000, 1999 - 2001
Irvine, CA
Work Experience
•
Associate Consultant McKinsey and Company,
New York Business Technology Office
•
Research Assistant Massachusetts Institute of Technology
Supervisor: Professor Troy Van Voorhis
Oct 2008 New York, NY
Nov 2003 - Aug 2008
Cambridge, MA
– computational simulations of electron movement in molecules
– troubleshooting complex computational systems
•
Research Assistant University of California, Berkeley
Supervisor: Professor Judith Klinman
–
•
May 2002 - May 2003
Berkeley, CA
theoretical and experimental work investigating quantum tunnelling in a soybean enzyme
Research Assistant University of California, Berkeley
Supervisor: Professor Paul Bartlett
Sep 2001 - May 2002
Berkeley, CA
– organic synthesis of a digestive enzyme inhibitor
Leadership Activities
•
2008 Battlecode Competition (6 / 196)
Massachusetts Institute of Technology
Jan 2008
Cambridge, MA
– annual artificial intelligence programming competition
consideration of multi-dimensional implementational limitations and rival
– strategies while steering development under tight time and resource constraints in a 4-person team that ultimately placed top 6 out of 196 teams
•
2007 Battlecode Competition (Top 16 / 157)
Massachusetts Institute of Technology
Jan 2007
Cambridge, MA
•
2007 MIT Taiwanese Career Fair (Organizer)
Massachusetts Institute of Technology
Nov 2006 - Feb 2007
Cambridge, MA
– designed and implemented online resume collection system
logistics, budgeting, and term negotiation with companies as part of a core
–
3-person team
– inaugural career fair of its series at MIT, attended by about 200 people
Publications
• Jeremy S. Evans, Chiao-Lun Cheng, and Troy Van Voorhis,
“Spin-charge separation in molecular wire conductance simulations”
Phys. Rev. B (submitted)
• Chiao-Lun Cheng, Qin Wu, and Troy Van Voorhis,
“Rydberg energies using excited state density functional theory”
J. Chem. Phys. (accepted)
• Qin Wu, Chiao-Lun Cheng and Troy Van Voorhis,
“Configuration interaction based on
constrained density functional theory: A multireference method”
J. Chem. Phys. 127, 164119 (2007)
• Chiao-Lun Cheng, Jeremy S. Evans, and Troy Van Voorhis,
“Simulating molecular conductance using real-time density functional theory”
Phys. Rev. B 74, 155112 (2006)
Languages
Natural Languages: English (native), Chinese/Mandarin (native), Taiwanese (fluent)
Computer Languages: Python, C++, Fortran 77, BASH, LATEX, Common Lisp